In Latin the suffix -que can be used to mean "and". For example:
Fames sitisque (Hunger and thirst)
Are there any modern Romance languages that use the suffix -que or something similar to it?
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This claims that Romanian şi (and) derives from "-que", but Wiktionary derives it from "sic" (thus). (Being Wiktionary, of course without any sources.) To me, going from thus to and seems like a big leap, but I don't know anything for sure about that. Edit: Apparently not that giant a leap, see comments.
As far as actual enclitics go, I'm not aware of any Romance language that uses one, but I only know about the 'big' ones (Italian, French, Castilian, Portuguese, Romanian).
Two French/Italian examples :
o French "quiconque" (anyone) < Latin quicumque (anyone)
quicumque = qui + cum + -que
Italian : chiunque
o French "chacun" (each one) < Vulgar Latin "cascunu" < Latin "quisque unus" (every one), maybe mixed with Ancient Greek "unum kata unum" (one by one).
quisque = quis + -que
If 'something similar' to Latin -que is good enough for you, present-day Spanish conmigo contains an old preposition or prepositional clitic suffix -cum hidden under the current from -go. [Our conmigo actually derives from cum mecum, where the -cum part of mecum was originally an adposition/a syntactic affix meaning 'with' arguably similar to the -que clitic meaning 'and'. That construction was, of course, redundant, a remnant of the stage at which late spoken Latin was becoming a head-complement language, what Romance languages soon were to develop into, but still contained important residues, mainly in VPs and PPs, of the earlier complement-head phase]. Less obviously, present-day Spanish quisqui, as in todo quisqui (= every body), cada quisqui (= each person), is a reflex of Latin quis-que (originally perhaps = somebody else).
Considering not the Latin form -que, but just a word meaning "and" that would somehow be present as a suffix, Romanian și appears as an etymological base in this way.
Without being an actual suffix that would work like in Latin in modern Romanian, the form și (and) from Latin sic (which gave yes in other Romance languages) may be considered to have acted as a suffix (maybe in Proto-Romanian?) that explains the etymology of a few words (conjunctions, determiners), although it doesn't act now as a suffix.
Totuși (however) < tot (all) + și
French has ouais in various senses of "yeah" (Collins Dictionary). Wartburg (FEW vol. 4 p. 444) derives it from hoc. See also language d'oc and language d'oil. Anyway, like hic, I guess that the coda is from a particle reconstructed with a voiced velar, possibly aspirated palatal (de Vaan: hic). This one is poorly understood. It is otter nonsense to equate ouais with -que, even when nequbi / necubi alternates the way it does, however.
The question concernes -que at the end of conjunctions. The major romance languages use a form of et "and" in enumeration instead. Fr. aussi "also, too" is from sic, so that's perhaps your best guess, analoguous to hic, hoc (hickery doo).
Ultimately it is not clear what the question is asking. As a matter of methodology, the vocabulary knowledge concerns the individual languages. The suffix being productive anywhere wouldn't really be a linguistic concern by the rules of this bord. As a matter of gaining a better understanding of the origins of the suffix, comparable to quisque, a broader view of Vulgar Latin is required. Latin.SE is over there, but evidence of spoken Latin is scarce, of course.