If I build the Builder's Chambers at the same time as someone builds his first wonder stage, should he be charged?
(I assume you mean Builders’ Union, where other players have to pay 1 coin to the bank for each Wonder stage they built.)
Yes, players have to pay also for stages built in the same turn.
From the Cities rules about Debts (page 4):
Important : monetary losses are always resolved at the end of a turn, after players have played their cards, possibly having paid for trade and/or obtained money...