You should be able to do this through the SE API. The question, answer, and post types have both a
downvoted field and an
upvoted field which, respectively, show if the requesting user has upvoted or downvoted the post. That information is considered private, so you will need to obtain and use an
access_token when making the request for the data on the posts.
I say "should be able to do this", because I have not actually tested that the data is valid, due to the difficulty in finding an appropriately-locked post which I know I've voted on in order to verify the
downvoted data is provided accurately. However, I'd consider it unlikely that a filter would have been put on the data provided by the SE API for the cases of a posts locked with specific types of locks.