Players do not need to discard immediately if they exceed the capacity limit during another player's turn. They only discard down to the capacity limit (9 coins, 9 cards, and 9 resource tokens) at the end of their own turn.

This is explicitly stated in the Capacity Limits section on page 11:

"If you gain an item that causes you to go over your capacity limit during another player’s turn, you do not need to discard down until the end of your next turn."

Summary

Players wait until the end of their own turn to discard excess items, even if they exceed the limit during another player's turn.

Source: Wyrmspan Rulebook (pp. 10, 11, 14, 2, 6)