There is no such rule about when in the season they need to play. As far as leagues go themselves, the League Operator can allow for X number of players to play throughout the season if they wish. The BCAPL only needs the Sanctioning $$$ of any player that plays a minimum of 1 time for the year.
It shouldn't matter when in the season they begin play.
In the league I run, we only don't allow any new players to play the final 3 weeks and playoffs if they didn't already play previously. This is done for 2 reasons. 1) So teams don't stack their team at the end to improve position or win playoffs with a team that didn't play all season, 2) To be able to track stat's, BCAPL Fees, verify players for Nationals, etc. It takes some time.