Origin of 'tada'

"Tada" is indeed an onomatopoeia (from OED):

Imitative of the sound of the musical flourish or fanfare (composed of one short note followed by one long note) which often accompanies an entrance, trick, etc., in various kinds of performance.

The word is imitating this classic horn sound.

Here is the first citation in 1913:

Coming front in utter disgust, he [sc. a conjuror] tells them [sc. the orchestra] that that won't do, that he wants something like ‘tadaa!’ from all of them. They seem to understand, so he goes off again. On his reappearance, however, he is met with a loud tumult, as all the orchestra shout out in unison the word ‘tadaa!’.