There are no Chick-fil-A's I would consider quick, easy and convienient to get to from Port Orleans.
In addition to taking more time to any of them, and risking traffic issues, Chick-fil-A can be a drive-thru traffic jam on its own.
I am the most familiar with the Wawa area I pointed out as we frequently go there when staying at Saratoga Springs.
If you really want to go to Chick-fil-A you can, of course, but I think there is a greater possibility it will take more time than you may like.
We like to save money and time with breakfast and other snacks, so we will usually buy some simple breakfast items and other snack items at the Winn-Dixie grocery story I mentioned in my previous post. We may go out to breakfast one other time, but that is about it.
A Disney, everything takes more time than you think, and to be able to get up and get going makes is much easier.