As far as Bingo goes, you need to check your local state laws. In many (maybe most) states it is considered gambling and requires a permit.
In Texas - as absurd as this sounds - you need to file an application for a permit for ANY game, even if there is no cost to play. The only exceptions are for a handful of people playing in a private home, nursing homes using bingo as a recreational activitity and maybe a couple of other things. But schools/parent groups would not be exempt from the rules (in our state).
When I talked to our county PTA unit, they said as long as we weren't charging any money we could have it on school property. I assume that if no money is involved that it is not considered gambling. I'm not sure if that helps.
That is a great idea. We could tie the Bingo night with the end of our Readathon and since the books are prizes that would be a great way to tie it together. In fact, I think people would be very inclined to attend because of the bingo aspect. We had discussed getting a guest author or librarian to help with the event after the Readathon and that might be able to part of the event. Thanks for the wonderful idea.
We don't really have a "book swap" but we have a family bingo night and each person participating brings in a gently used book to use as Bingo prizes. We have done this for years and many people clean off their book shelves and bring in boxes of books for the kids. We borrow the bingo equipment from a local company. So this event is low cost and kids love it. Several parents have asked if next year we can swap books for parents.
Hope this helps!