Many, including myself, have pulled and driven big rigs right to the outskirts of Key West. I would't go into town though.
The Navy campground is at about Mile Marker 2. That is 2 miles from the end of the Overseas Highway, US 1. We have pulled our TT and driven our Class A pulling a Toad to that campground with no problems.
Prepare your route carefully. Real Estate is VALUABLE, so turn around spots are few. Likewise, gas stations are tight, so gas up on the mainland. Gas, as everything else is more expensive in the Keys.