There are several Residence Inns close and they accept dogs. We stayed at the one by SW and convention center. Plus they have kitchens and hit breakfast.
The Drury Inn in Orlando takes dogs. We were just there a couple of weeks ago and stayed for a week. It's a very nice hotel. They have several places you can walk your dog and they provide bags for picking up the waste.
I have no idea about "cheap" as we were there over Christmas and a suite (sleeps four) was about $150 per night. Off-season though might be less expensive.
They also provide a hot breakfast every morning and at 5:30 each night they have something called "kickback". Finger foods, chips, baked potatoes, mac and cheese, hot dogs, sodas, coffee, etc.
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