Can you take in a cooler for a picnic?

Yes you can

You can bring a backpack with food. I'm not sure about a cooler.

We take a soft cooler with us, with sandwiches, juice and water bottles.

I took one before and they said i have to carry it e every where i went in the zoo they don't like when i pulled it it has wheels too so maybe you have to rent out a wagon

And theres also a big park in the zoo for picnics..

I haven't seen anyone bring an ice chest before. Maybe they only allow schools to bring them in. Not quite sure. But you can always just bring something in your purse or back pack. They don't check you

There's picnic areas outside for personal food and they sell food inside at food courts, but I didn't see anyone with an ice chest within the zoo itself.