I do not believe it is possible
I don't think so
There's a sign at entry that says no outside food or drink. I had my own reusable water bottle and no one said anything. The cafe there is only open until 3pm. There are benches and such around but nowhere to really lay a blanket down in my opinion (also there are gators so you wouldn't catch me sitting that close to the water anyway). Also masks are required in the gardens at the moment. As you drive in though there are large grassy areas, I'm sure you could picnic over there no problem.
I have not seen people bringing their own snacks. However there is a Cafe inside which has delicious food.
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