관객을 and 관객이 are both correct, though I think 관객을 is more natural here.
This is a construction that corresponds to English "ask/tell/cause (someone) to do something": here, Korean usually allows both subject and object markers in the position, although in some cases one of them might be more natural than the other.
Sometimes "-에게" is also allowed (e.g., "to tell (someone) to...").
선생님은 철수를/철수에게 앞으로 나오라고 했다. = The teacher told Chulsoo to come forward.
선생님은 철수가 앞으로 나오라고 했다. = (same, though it feels like the teacher emphasized none other than Chulsoo should come forward.)
선생님은 철수가/철수에게 칠판을 지우게 했다. = The teacher made Chulsoo erase the board.
선생님은 철수를 칠판을 지우게 했다. = (same but less natural, probably because we have two consecutive -를?)
그는 그 프로젝트가/프로젝트를 실패하도록 방치했다. = He abandoned the project to fail.
Back to the sentence in the question, the hidden subject is clearly 배우들 (actors). So,
연극 중간에 [(배우들이) 관객을 무대로 나오라고 할] 때가 많았거든.
= During the play, there were many times when [the actors asked the audience to come up to the stage].
= During the play, the actors would keep asking the audience to come up to the stage.