Absolutely rent a car if you can, especially with that many young children. Your hotel will most likely share a shuttle with other hotels, and they can get extremely crowded. Who's going to stand? Also, the shuttle times may not be convenient for you (or frequent enough). And finally, it can take upwards of 45-60 minutes given these factors to make it to the parks, only to be dropped off a long distance from the entrance. So you've got to schlep all the little ones all that way just to even get to the turnstyles.
With a car, the advantages are:
No shuttle fees to/ from the airport
You can come & go at your convenience, and everyone will get a seat
Quick drive directly to the park of your choice, whenever you want to get there
Much faster/ easier to get to other resorts (like Wilderness Lodge) than using Disney transportation
Easy access to TONS of off-site restaurants, which can save you big $$$
You can come & go as you please, just save your parking receipt. Disney only makes you pay once
My advice for finding better prices on rentals is to join the "insider" clubs for the rental companies (we've gotten great deals from alamo) and check out mousesavers.com for current coupon codes. If you've got a Costco, BJ's,
AAA, Visa, airlines, or other membership they all provide coupon codes as well. I've made/ cancelled/ rebooked a dozen or more rentals without penalty when a new coupon code comes along. We like Alamo because you can stack discounts.