I don't know if any of the resorts let you just walk up to one of the bodies of water and start fishing.
I do know that you can set up fishing excursions through WDW Recreation.
I did go "cane pole fishing" at the Port Orleans Riverside. They have a designated fishing hole. My DD was 10 at the time and she enjoyed it. I don't believe it was to costly. Of course, you release the fish after you catch it.
Good luck.