You can, you do have to ask permissions, so that you're not over booked. Not sure abt bringing grill. There are some grills in the area.
You have to call and book a date ahead and you I'm sure you can always bring your own grill although the have about 2 permanent grills per booth....and ofcourse clean up after yourself... Feel free to ask me if you need Photo coverage for any events.
Tammy is right, call ahead to make sure someone hasn't reserved a space. There are grills available to use near the pavilions.
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