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 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).

  [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