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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 timeline is not provided in the API.

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