Some folks do indeed rent goats for brush clearing. Obviously, it's a very local thing, so you might want to start asking around at feed stores, the cooperative extension office, craigslist, or your neighbors. If you have a livestock vet in your area, they may know of someone.
For those looking to rent (or even buy) goats or sheep I'd highly recommend checking Facebook or other social media for local groups. I know for my area there's at least 3 poultry related groups, and at least one (probably 2) goat/sheep group on FB. Since travelling long distances usually isn't viable for these things something in your county or a neighboring county is usually required to make this work.
The Greenhouse of the Future ebook by Francis Gendron