Personally since you are the only adult I would try to stay onsite and utilize the Magical express. Then you wouldn't have to worry about lugging around luggage and navigating three children through the car rental. Can you rent through Enterprise? They may even bring the car to you hotel (not sure about this).
I'd definitely feel better with a car, in case we need to run to Walgreens or visit an urgent care. It would be our first big vacation together so I'd have a little nervousness and would want to have a car! But, I know through my CC points that I have to get the car from the airport; they only rent that way.
I was looking at doing a quick, cheap trip last year and found Last Minute Travel. They had great rates for the Hilton in DTD. I think it was $84/nt. That was a regular hotel room though. I didn't end up booking through them so I can't recommend them. I actually found the site through a few threads here. If you do a search you can probably find some reviews from people who have actually used them and the tricks for finding out the name of the hotel before you book. The Hilton in DTD participates in EMH.
That's good to know that the Hilton gets EMH. Where exactly do you find out which hotels are DTD hotels? Is it by address, or is it named something special when you do a search on Orbitz or Travelocity or...??
Bonnet Creek adds a "guest confirmation fee" for less than 7 days and most Home rentals (Windsor Hills add a $75+ cleaning fee for most rentals)
Would you consider a home rental? If so..you might be able to find a renter that has a "hole" in their calendar/not a complete week available...and sometimes that have some great deals!

When are you thinking of going?
I'm not sure about the whole "home rental" thing - it concerns me that a big house may not be properly cleaned after every guest. You have more of a chance of a clean place with a hotel room, I think! I'm not against it yet, though. More research must be done.
We stayed in a 3 bedroom condo at Emerald Island in 2010 and are staying here again in July. EI is a gated community, 10-15 minutes away from all the parks and this unit is very close to the clubhouse/pool. It was only $500 for the week and two of the bedrooms had twin bedrooms(perfect for the kids)
http://www.vacationhomerentals.com/vacation-rentals/Kissimmee-Florida-vacation-rental-townhouse-proID-42665.html
For souvenirs, I have always given my son a set budget. Once it's gone, it's gone. I encourage him to "window shop" and take pictures of things he likes and then decide what he wants later in the trip.If any of the kids have birthdays between now and the trip, they can also let people know they would love Disney gift cards as presents.
Thanks for the info; I'll look into Emerald Isle. I figured that if the kids are really going to go to Disney for the first time, there will be a lot of grandparents who want to give a little money to them to spend. They'll have more than enough for a few souvenirs! Both of the boys have birthdays in late summer, but the girl just had a birthday. And regardless, I'll get them something too! I thought I'd encourage them to either take a photo (like you said) or write it down (I'll carry a small notebook anyway, so they could use that).
I second using Skyauction.com. You do not have to stay the full week to get that rate. We've used them several times and have always been pleased. Our favorite place when we can get it is Summer Bay. I've seen Bonnet Creek on there often too, but we haven't stayed there yet.
Another place to check is ebay. 2 years ago we stayed at Holiday Inn Orange Lake Resort for about 500 for the week. That was for a 2 bedroom villa and the kids LOVED the waterpark there.
Now, on skyauction.com I saw that they say you have to arrive on a certain date. I wouldn't be arriving then; would that matter? For example, if it says you have to arrive on a Saturday but we don't show until Monday, will we be SOL?
Also, are these refundable? If I tried to jump the gun and grab those prices but then we weren't able to go, could it be refunded? I'll have to go look at the fine print!
We are staying at Bonnet Creek in April. I rented a three bedroom for $110 a night including all taxes from 5starresorts. I believe the two bedroom was $99 a night. That would be less than an onsite hotel I would think.
Wow, it would be nice to get it for that cheap.
It would be nice if the planning were a little easier! I wish it wasn't so mind-numbing to try and get a good deal. I had a great, quick trip in January and trying to figure out a budget summer trip is the complete opposite of that experience!!