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Arabic languages include only three vowels: /a/, /i/ and /u/.
Japanese is the only language I know about that doesn't have a /u/ sound - it has /ɯ/ instead.
Do there exist any languages that do not use /a/ or /i/ or they could be considered universal vowels?
It depends on what you mean by "/a/", "/i/". First, slash brackets refer ambiguously to "phonemes" or "underlying forms". Only phonetic forms, notated with square brackets, have directly-observable phonetic properties. A propos that point, there is a high back vowel phoneme in Japanese, which is pronounced more like [ɯ] than like [u], and on that grounds you might claim that Japanese has no /u/ but it does have /ɯ/. The principle here would be that the underlying / phonemic value of a segment is the same as (one of) it's surface values. On that basis, we would have to conclude that Norwegian does not have /u/, because if you compare the pronunciation of French "tout" and Norwegian "to", the vowel of Norwegian is /ʊ/, not /u/ -- the vowel typically transcribed in IPA as [u] is noticeably lower than the cardinal vowel standard. The problem then is that language descriptions typically leave out that level of phonetic detail, so you often can't tell if supposed "u" or "i" are pronounced as [u, i] or [ʊ, ɪ] or something else. With that caveat, Somali Zigua has the vowels [a ɛ ɔ ɪ ʊ], and not [i, u].
Answering the question for "/a/" is even more difficult. The vowel "a" of Arabic is distinctly different from the vowel "a" of North American English (as in the word "cot"). The Arabic vowel is rather close to the standard cardinal vowel value of "a", so under any interpretation, Arabic has /a/. English, at least certain dialects, do not have that vowel (though in some New England dialects it exists in words like "park"). In fact, there is no distinct IPA symbol for notating the low central vowel of "cot", and language descriptions typically use the letter "a" in case there is only one low unrounded vowel, so it is only when you have a contrast like in Norwegian that people are forced to distinguish "a" and "æ", or "ɑ" and "a".
In some west coast dialects (e.g. in Seattle) the vowel of "cot" is the same as the vowel of "caught" and furthermore the vowel is phonetically more like [ɔ] than it is like canonical [a]. If by "a" you mean a low unrounded vowel, then I would say that Enumclaw English is a language without /a/ (I pick Enumclaw because there is an urban / rural distinction that complicates the matter).
There is a classical analysis of Kabardian by Kuipers where the language does not have /i/, or many other vowels, although phonetically the language has a huge number of vowels. This would be an instance of economy-driven phoneme-elimination, for the sake of reducing the vowel inventory to just one phoneme. There is a similar many-to-few reduction of vocoids in Marshallese, which has been analysed as having just a 4-height distinction between /ɨ ɘ ɜ a/, where the 12 surface vowels derive by phonetic interaction between these 4 vowels plus palatalization and labialization on neighboring consonants. Under that interpretation, Marshallese has no /i/. Mark Hale has argued cogently that it is inappropriate to claim that vowels have any specific values of rounding or fronting in the phonology.
It's been claimed that Margi (Chadic branch of the Afro-Asiatic language family, Nigeria) has only two vowels, /ɨ/ and /a/.
There have also been sporadic claims of other languages with small or anomalous vowel systems, for example Iatmul (Ndu language family, Papua New Guinea) has been described variously as having: one (/a/), two (/ə/ and /a/), three (/ɨ/, /ə/ and /a/), seven and twelve vowel phonemes. So by at least some analyses Iatmul has no /i/ phoneme.
So the answer may depend on how flexible /i/ and /a/ are allowed to be. It may be better to recast the question in terms of features, front/central/back and high/mid/low and ask if there are languages with no (e.g.) high vowels, no low vowels, only mid vowels, etc.
Nearly all languages have at least three phonemic vowels, usually /i/, /a/, /u/ as in Classical Arabic and Inuktitut (or /ɐ/, /ɪ/, /ʊ/ as in Quechua), though Adyghe and many Sepik languages have a vertical vowel system of /ɨ/, /ə/, /a/. Very few languages have fewer, though some Arrernte, Circassian, Ndu languages have been argued to have just two, /ə/ and /a/, with [ɨ] being epenthetic.
It can also be said that English has no [a] because typically it has only [æ], [ɑ] and [ɒ]
One of the most common vowels is [a̠]; it is nearly universal for a language to have at least one open vowel, though most dialects of English have an [æ] and a [ɑ]—and often an [ɒ], all open vowels—but no central [a]. Some Tagalog and Cebuano speakers have [ɐ] rather than [a], and Dhangu Yolngu is described as having /ɪ ɐ ʊ/, without any peripheral vowels.
Some other languages without [i] or [u]
[i] is also extremely common, though Tehuelche has just the vowels /e a o/ with no close vowels. The third vowel of Arabic-type three-vowel system, /u/, is considerably less common. A large fraction of the languages of North America happen to have a four-vowel system without /u/: /i, e, a, o/; Aztec is an example.
You write: “Japanese is the only language I know about that doesn't have a /u/ sound - it has /ɯ/ instead.”
Although you are using slashes, I think you are really talking about phonetics (“a /u/ sound”) and not about phonology. Phonology is an abstraction. There is no overwhelming reason why you should not analyse the Japanese [ɯ] as phonological /u/, nor is there any good reason not to analyse English [æ] as phonological /a/. These are just abstract symbols, like the symbols used in algebra. It would be better to ask whether there are any languages that have fewer than three vowel phonemes. Your question, as it stands, in unanswerable.
I don't have much knowledge on the language, but the Arapaho language is the only language I have ever seen without any low (/a/) vowels. There is still a four vowel system, but whereas Central Algonquian languages like Ojibwe and Cree have /a e i o/, Arapaho has /ɪ ʊ ɛ ɔ/.
Interestingly, there's pretty good evidence that Proto-Indoeuropean didn't have any kind of /a/ sound. The /a/ sounds in descendent languages all seem to have come about through consonants low-colouring their phonetic surroundings.
Proto-Indo-European is widely believed (see e.g. Beekes) to have only /e/ and /o/ (and their leghtened variants). This is somewhat a simplification because u̯ and i̯ which behave as consonants morphologically, could be vocalized into vowels between consonants. So it is better to just say that PIE had only /e/ and /o/ kernel vowels.
This may be not the case for the most older stage of PIE, the so-called pre-PIE. In that language it is quite possible that the only two vowels were /e/ and /o/ and possibly even only /e/ because /o/ appeared as unstressed /e/.
For instance, the traditional reconstruction for the word for daughter is dhuga̯tēr which has vocalized /u/, but according to Kloekhorst the most ancient form was dhu̯ega̯tr.
Iranian Persian and Tajik have lost straight /a/, and you can make the case that some English and some German dialects have too.
See: Persian phonology and its historical shifts
It is surely the case for many more languages, but asking us or reading eg https://en.wiktionary.org/wiki/Category:Alemannic_German_terms_with_IPA_pronunciation does not scale. The more interesting question is: What sites or tools exist to search over IPA pronunciations of an entire languages or many languages?
