Obviously, vote time data is sensitive and [censored to protect voter anonymity][1].

So, scanning [the list of available objects in the API][2], we see that:

 1. A **user's downvote** times can be obtained via [the `/users/{id}/reputation-history/full` route][3] 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).

  [1]: https://meta.stackexchange.com/questions/933/why-is-vote-time-missing-in-the-se-data-dump-and-sede
  [2]: https://stackapps.com/a/7686/7653
  [3]: https://api.stackexchange.com/docs/full-reputation-history
  [4]: https://api.stackexchange.com/docs/questions-timeline#pagesize=100&ids=6091&filter=!6JE_AUixqf1)D&site=stackapps&run=true