I went to Tokyo quite a few years back as a single traveler on a budget, and recently returned with a family of four and a little more disposable income. I have to say, there are no hotels right near Disney that I'd consider budget, especially for a single traveller. You'd be hard pressed to find anything for under about $200 US, however, they are very nice hotels if you want a splurge.
When I tackled Tokyo on my own, I found the youth hostels in downtown Tokyo to be very good value and very clean and safe. The Tokyo Metro is amazing; once you've figured it out it's only about 45 minutes to Disney and costs about $5. If you could leave your luggage at your accommodation downtown it would be fairly quick and easy.