First, I would tell them that - just for them, because I'm a nice guy who wants to be as supportive of their efforts as possible, and because it's a large function with significant sales potential - I will forgo my "usual" $500 a day event shooting fee. Heck, I might even make that $750 or $1000! Doesn't really matter, does it, since I'm forgoing the fee!
But, no, I won't pay out a percentage of sales.
Now, I might pay a flat vendor fee for access to a really big, well established and very high sales potential event, if I were given exclusivity.
And, I will support a non-profit organization in many ways that I feel are appropriate; such as donating limited, non-exclusive usage of some of my photos for their website, newsletters, programs, brochures, etc. In return I expect to get a link from their website, credit and a listing in their newsletter, programs and brochures or wherever images are used, etc.
Perhaps I'd even donate limited image usage for some fund raising efforts, such as a calendar that's offered for sale, produced at their expense and with all proceeds going to them. Or - in a situation I'm currently discussing - postcards to be sold for fund raising in a similar manner.
I probably would not be quite as generous with any for profit organization. But I will support them in some ways, such as limited use of my images on their website in exchange for a link.
I would also consider a sponsorship, with either profit or non-profit, such as a donation of prizes (usually prints or print packages, something related to my work that I can buy wholesale). For my sponsorship, I expect to receive a high level of promotion in return.
Oh, and for most events with some reasonable potential for sales, I don't actually charge a shooting fee. I shoot "on spec", which is plenty of risk to take in my opinion.
In either case - for profit or non profit - I will not pay a percentage of my sales, period. Doing so would just set a very bad precedent, I feel. It shouldn't be about money. A percentage of sales sounds way too much like a "kickback", which sounds unprofessional to me, plus it requires me to open my sales records to them. And, it becomes an ongoing bookkeeping nightmare that might last for years into the future (I just recently sold a set of images from a 2006 event!).
If they insisted, I'd just say "Thanks, but no thanks." and walk away.
A note regarding exclusivity... This is not just greed on the part of the photographer, although most people assume that when you first bring it up. It's also a key safety issue. The last thing a sports or event organizer needs is several photographers tripping over each other and getting in the way of their participants, vying to get "the shot"! Multiple sources also confuse their participants, regarding where to look for photos afterward, how to order them, prices of prints, etc., etc.
From my perspective, though, unless it's a huge event, a non-exclusive gig is simply not going to be profitable enough to make worth my while. So, I've walked away from several where we were denied exclusivity. On two occasions, the organizers eventually hired us for events... And gave us exclusivity.
p.s. I have donated a 10% tithe to a church, out of the sales made while shooting an event there. That's a quite different matter, in my opinion.