Going to repost this here, as I realized it was because we’ve already completed the goals.
I stand corrected! According to the official New Feature: Crews Q&A, you just have to be part of the crew and participate in the active event to be eligible for the rewards, and as long as you contribute a little after it starts, you are good to go.
My test was based on joining after an event’s goals had already been reached while the event was still active - since we usually finish the previous and current ones on the first day. Because you can no longer contribute once the max goal is hit (for instance, once all 1,000 successful parties are completed and extra parties stop adding to the total), it seems like anyone who joins after that point misses out.