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Obviously, vote time data is sensitive and censored to protect voter anonymity.

So, scanning the list of available objects in the API, we see that:

1. A user's downvote times can be obtained via the /users/{id}/reputation-history/full route for the authenticated user only.
   (Even for downvotes that cost no reputation)

2. There is no object that returns a user's upvote times.
   Currently, your only recourse is to log into the website and scrape the
     https://stackapps.com/users/current?tab=votes&sort=upvote
   pages.

3. Question up/down vote times, normalized and aggregated to midnight can be had via [the /questions/{ids}/timeline route][4].

4. Answer up/down vote times are only available as a subset of the parent question's timeline.

5. You cannot see voting time information for anybody but yourself (or the authenticated user), via any means (Website, API, SEDE, Data Dump).

[4]: https://api.stackexchange.com/docs/questions-timeline#pagesize=100&ids=6091&filter=!6JE_AUixqf1)D&site=stackapps&run=true