Can a runner steal bases while batters are switching?

A runner is allowed to advance at their own peril whenever the ball is in play. The ball is not automatically dead after completion of a play. But if "time" has been called, then no advance is possible.

Usually, the pitcher will control the ball and be standing on the mound while the new batter enters, so advancing is unlikely. If the defense wants to discuss the batter or have a conference, they will request "time".

So in many cases, the answer is "yes", the runner could try to steal. In some other cases, time has been called and the runner must wait for for the ump to call "play" before a steal would be possible.

5.12 in the MLB Rulebook covers situations where the umpire may call "time". Completion of a play and end of an inning are not listed.

Wikipedia has a summary of the dead ball and time rules, and has a further description:

In general, the ball does not automatically become dead after playing action ends. So, for example, although the recording of a third out generally winds down a half inning, the ball is not automatically dead. If it is to the advantage of the defense to attempt to record a fourth out for any reason, the ball is live and such a play is permitted.