I've seen teams grilling away from the field. If your plan is to tailgate in the parking lot, there's plenty of places to go. If you want to grill by the field, that might be a problem with the organizers.
I would doubt it, but I can't find any facility rules on the website anywhere. I know they don't dogs at the fields, feel like I've seen it in print, but can't find that anywhere either. There's lots of space surrounding the complex, like adjacent parking lots that are only in use during the work day. Could probably sneak over to one of those.
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