Are there languages where articles appear—as independent words—on the right-hand side of the noun phrases they occur in - in other words after the head noun in the noun phrase?
One problem is determining that the item is an article, not a demonstrative (assuming that we use semantic tests and not conventional translations into English to decide that matter). There might be some question as to the obligatoriness of the article, if one believes that "if you have articles, everything is either marked with a definite article or an indefinite one", but this doesn't even pass the empirical sniff test for English (Sentences with no articles exist).
The second is determining that the thing is a word and not a suffix. The distinction is elusive or nonexistent in some grammatical frameworks. A semi-test – which arguably fails to capture the word / affix distinction – would be whether the item appears after non-nouns in case the NP has a structure like N+Adj. This is not a particularly good test, because not all affixes go on a single part of speech (like "on the noun"). A good example of that is the English possessive marker -'s, which goes at the end of the NP. This has been termed "edge inflection" in work on the topic, by Philip Miller, where there is a syntactic requirement that the constituent have a certain feature and the value travels down the right edge of that constituent. Thus in "The King of England's name", the possessive marker is realized on the last word of the NP that is the possessor, and can be indefinitely separated from the head N.
The point is that "appears at the edge" is not a sufficient diagnostic of wordhood. We can call on certain other facts to make an ad hoc argument that 's is a suffix and not a word in English. The most compelling is the allomorphy argument, that possessive 's does not combine with plural -s, though it does combine with plurally-inflected nouns ("the children's dog"). There are other variable issue regarding realization of 's on "Jesus" (some people say "In Jesus name" rather than "In Jesus's name"). A second is the basic phonological fact that every word in English has to have at least a vowel. So the main question for examples like Persian is what the arguments are for the thing being an independent word rather than an edge-inflected affix. This page discusses what they call the Persian indefinite enclitic (a nice thing about that page is that you can click on (many) examples and hear them, and not have to be able to read Persian -- unfortunately, some of the examples are misordered); this article says "There are no dedicated definite or indefinite articles in Persian", but also calls -i an enclitic (doesn't deny its existence, rather it is a terminological matter). -i has a certain degree of allomorphic selection: variant ye after the plural suffix; it seems (Perry says) that the ezafe marker is dropped when the indefinite clitic appears on the head noun. I can't say that I've seen a linguistic article explicitly comparing the clitic vs. full word question for the Persian indefinite marker.
A candidate for independent post-nominal word exists Logoori. There is a lexeme -éné which appears towards the right edge of the NP, which can be preceded by any part of speech (not just nouns), which seems to signal definiteness, and passes certain tests of word-hood. Some examples are:
kémóórí !chééné "the calf"
kémóórí kɪ́rítú !chééné "the heavy calf"
kémóó!rí cháMáróvá chééné "the heavy calf of Marova"
vímóó!rí vyáMáróvá vyééné "the heavy calves of Marova"
kémóórí chééné vʊza "only the calf"
The NP is head-initial (I use "NP" non-technically), and the purported article is at the right side of the NP, though can be followed by vʊza "only". As a first approximation, it agrees in noun class with the head noun (hence the chééné ~ vyééné variants). Actually, when nouns simultaneously have two classes (as is the case with the locative classes) agreement can be with the lexical class of the noun, or the locative class, hence hányʊ́ʊ́!mbá yééné ~ hányʊ́ʊ́!mbá hééné "at the house".
There is a morphological argument that -éné is not a suffix. The specific class morpheme that is selected under agreement is determined by the formative that agreement attaches to. There is a distinction (in some classes) where the class marker has one shape when attached to nouns and adjectives, and another shape when attached to possessives, demonstratives, or numerals. Thus you get ma-rwá! má-rá!hí g-ɪ́ɪ́tʊ ga-vɪrɪ́ "our two good beers" (beers good our two), with the noun and adjective exhibiting the primary class marker /ma/, and the possessive and numeral exhibiting the secondary agreement morpheme /ga/. In ma-rwá má-rá!hí g-ééné "the good beers", the article, which follows the adjective, has the secondary marker /ga/, not the primary marker /ma/ (seen on the adjective itself). Accounting for this would be challenging if -éné were a suffix on the adjective.
Finally there is the question of the semantic function. There are a number of distinct demonstratives in the language, such as /yI-AGR/ "this (visible)", /AGR-nʊ/ "this (possibly not seen)", /yI-AGR-o/ "that (relatively close)" and /AGR-ra/ "that (very remote)", which express notions of physical proximity. /-éné/ on the other hand signals that the referent is known from the discourse, and functions to tell the interlocutor that the thing in question is known from the discourse.
The biggest impediment that I see to answering the question is distinguishing "article" on functional grounds from similar kinds of words.
In Persian the indefinite article /i/ can be attached to a noun, or to a noun+adjective phrase. For example:
pesar-i “a boy”
pesar-e bozorg-i “a big boy”.
Though of course in written Persian this article is written to the left of the head word. Just as a comment on the inadequacy of this "left / right" terminology.
The answer to your question definitely seems to be "yes".
However, finding clear-cut examples has been difficult for me (though I would imagine that there are a number of them). It is of course often difficult to draw a line between independent words and bound clitics, or between articles and demonstratives. Among the well-known European languages, languages with postposed definite markers occur in two main areas that I know of, but the difficult point is the "independent word" requirement. It seems that many argue that the Balkan-sprachbund languages have definite suffixes, not independent words (see "Romanian definite articles are not clitics", Albert Ortmann & Alexandra Popescu). (There was another question on this site about that: Why is the definite article in Balkan languages always called a suffix when it really seems to be part of the inflection?) The status of the articles in North Germanic languages seem to be similarly dubious. It seems that, for examples of postposed definite articles that are obviously independent words, we have to look at lesser-known languages.
I would tentatively propose Miskito as a clear example of a language with a post-posed definite article that is an independent word; Wikipedia gives the example aras ba 'the horse'. Interestingly, in this language demonstratives precede the noun phrase, and it appears demonstratives and the article can co-occur, with the latter marking the end of a relative clause (as the article comes at the end of the noun phrase). The following example comes from Definiteness, by Christopher Lyons (p. 62):
baha waikna naiwa balan ba baku win. that man came today REL said so. 'The man that came today said so.'
The rest of that page and the following few pages also seem to have a lot of relevant information, so I would encourage checking it out (I don't know how to summarize it, and I don't want to quote paragraphs upon paragraphs).
Another, less relevant source that I found that has some relevant, maybe-interesting info and references:
"The order of noun and demonstrative in Bantu", Mark Van de Velde, mentions that in some Bantu languages, there is apparently (based on descriptions of their grammar) a tendency for the same morpheme to occur pre-posed when used more like a definite article, and post-posed when used more like a demonstrative.
As I see it, the preposed demonstratives are on a path of grammaticalisation from textual anaphora markers to definite articles. Why this evolution affects their position vis-à-vis their head noun is not clear at this point. Dryer (1992: 104) notes that there is a statistically significant tendency for articles to precede the noun in VO-languages. Curiously, the languages of Africa form an exception to this generalisation.
I found Miskito by looking through the WALS map showing the overlap between languages with a definite article distinct from the demonstrative, and OV word order.
The answer to your question is not clear. What an article is and what separate is is arguable. (many of my references are from WALS)
Historically, the definite article derives from demonstratives, explicit adjectives that mark proximity, and indefinites, a vague singular, derive from the numeral for one. There are many languages in Africa where demonstratives and numbers come after the noun. Among IE, the Celtic languages seem to be the only ones with post-noun demonstratives (the indefinites come before the noun in Celtic). WALS does not have a chapter specifically on articles, therefore the difficulty in saying definitively yes or no.
As to things called articles specifically, again there are many languages that have them after the noun. However they all seem to be inseparable, that is, they seem to be suffix-like rather than separate words.
frate - brother
fratele - the brother
fratele mare - the big brother
I can't seem to find a language where there is an article that comes after a noun -and- is obviously separable. (as an aside, it is rare that an article comes before the noun and is not separable).