In my vanilla survival world (which is in hard difficulty), I have killed many vindicators, but they only seem to be dropping Sharpness I axes. All of these vindicators were encountered during a raid; I have not yet been to a woodland mansion.

I have killed them with an axe or a sword without looting, sometimes with Looting II or III.

Seems to not affect vindicators that are spawned from a spawn egg.