Yes, you can use Uber! I actually plan on using Lyft when I go in August because you get $10 off your first 5 rides when you sign up, and I did fare estimates to see I'll never really spend more than $10 going to/from All Star Sports and my ADRs. This is the discussion that talks all about Uber/Lyft at WDW:
http://www.disboards.com/threads/everthing-you-wanted-to-know-about-uber-lyft-at-wdw.3443634/
Warning: I assume your kids are in car seats? I'm honestly not sure how all of that works with Uber/Lyft. The post I linked talks about it a little but I don't know any details.
As for Amazon delivering to Disney, yes they do. BUT I looked into Prime Pantry for my trip, and realized it is NOT a 2 day delivery window like regular prime. Its a range of days so you wouldn't know when your food is actually arriving. I guess you could order food with regular Amazon Prime, but I don't think it's necessary for you. Since your family will have so many free bags with Southwest I think you are better off just packing a ton of food. I always do that and you'd be amazed how much food and heavy drinks fit in a 50b bag, and I just weigh them on a regular scale. Another thing: If your family is going to have a lot of bags, that's another reason to stick with Magical Express since Disney takes your bags from the airport right to your room. That way you can save the time of baggage claim and lugging bags all over the place.
The fact that your rental is only $80 did get my gears turning a little bit. I just feel as if overall there are more pros overall to using busses and Magical Express (as well as Uber/Lyft) so you don't feel so overwhelmed. Also, neither a car nor a bus with be a quick escape back to the room and they do both involve a little bit of walking- busses are a little less though. Just pack a bunch of food and let Disney basically do everything for you in regards to transportation! Honestly, it is going to be hot, tiring, and there will be cranky kids at some points whether you're riding on a bus or in a car, so I would stay stick with busses to eliminate the aggravation of driving/navigating. That is my final thought and I am posting before I start weighing the options again!