Question: We are in the middle of the Charleston and North (player 4) realizes they forgot to take their 13th tile. Can they go ahead and take their 13th tile?

Answer: Yes, they can pick their thirteenth tile but only before the Charleston or before East has discarded their first tile. 

Source:  Mah Jongg Made Easy 2024 page 15 OTHER TIPS AND RULES FOR THE CHARLESTON, bullet #1