Question pretty much says it all, I'm curious as to whether every language uses/requires auxiliary verbs to express tense/modality/aspect etc.? Thanks.
The precise answer to that question will depend highly on what is the precise meaning intended for auxiliary verb. If you insist that an auxiliary verb is a separate word which expresses a precise grammatical property, then Japanese is a likely candidate. In this language, tense, mood and modality are expressed by a specific ending attached to the stem-verb. None of these endings may live a separate existence (see here for a reference), so it would seem very strange to call them auxiliary verbs.
On the other hand, as the same reference indicates, there are verbs in Japanese which have an independent meaning but can also attach to a stem-verb to convey a nuance, as in
食べきる (eat it all) vs 食べる (eat)
where the verb きる which ordinarily means to cut has been appended to the verb to convey the idea that the action was brought to full completion. Just as one would normally not consider finish in
I finished reading.
as an auxiliary in English, I would not consider these Japanese constructions to be real examples of auxiliaries (in the traditional sense of English grammar).
I'm guessing that using a restrictive definition of auxiliary, there are many more examples, for instance in synthetic languages.