Can you call for a review of a play to see if there is a penalty in the NFL?

No. You cannot challenge a judgment call by a referee and a penalty is a judgment call.

However, if during the review the officials determine that there were 12 players on the field at the time of the play, that penalty can be administered. But a coach cannot officially call for a review due to his thinking that the referees missed a penalty on the play.

Actually, penalties such as illegal forward pass, forward pass beyond the line of scrimage, and touching of the ball by an ineligable receiver are reviewable. So penaties that are factual are reviewable.