Danielle --
Just to follow up on my above post and to confirm -- in addition to the other places I mentioned (Earl of Sandwich, Napolini and Jazz Kitchen Express) and the other suggestions that have been made in this thread, La Brea Bakery Express (next to La Brea Bakery) does, indeed, offer food to go as well -
https://disneyland.disney.go.com/dining/downtown-disney-district/la-brea-bakery-express/.
For some reason I was assuming that you were not planning to call first and then pick it up -- though maybe you were? When I read your initial post I immediately assumed that you would just walk up to one of the DTD places that have been mentioned, place your order and take it to go.
In any case, now you know you have several options and can dine by the pool!