Can it be done in Data Explorer, or would I need the API for that?
I was thinking about how to write a query for the Sportsmanship badge, and I stumble on this. I can see that there might be privacy issues.
There is no way for a lay user to see what you have voted on. Neither the API nor the data explorer allow it.
Mods can see some voting info, and the devs can see it all, but that's it.