Yes. In the poll's Voters tab, you will be able to search for any voter, and see the history of that voter's participation in that poll - including the date and time of a voter’s cast ballot. (However, no one can tell how the voter voted, unless the question is clearly marked on the ballot as non-anonymous.)
In addition, when a ballot is cast, BallotBox sends a Ballot Recorded email to the voter. If the voter did not see that email, it could be in the voter's spam (junk mail) folder, or otherwise blocked by a spam filter.