OK, on the '37, the bed is the same width as the rear most part of the frame. As the frame narrows toward the front, it is close to 2" narrower at the front spring hanger of the rear springs then the bed sides.
I am using a 1979 Trans Am/Camaro 10 bolt rear axle with stock 15x7 Chevy Ralley wheels. I believe the wheels have 3 1/2" offset toward the front, most common wheel available. With that in mind, I have 2 1/2" clearance on the inside, tire to bed side, and 1 1/4" to the outer lip of the fender with a 235/75R15 radial at 25psi. And again, my fenders are 2" wider then stock. With stock width fenders, my tires would not fit with this axle or wheel offset. From bed side to outer edge of tire is 13 1/2" and from frame to outer tire is 15 1/2" in front and 13 1/2" at the rear.
Tire to top of fender is about 9" with full tank of fuel and no load in the bed with three out of the 9 springs removed.
If you run a straight edge parallel to the rear most frame rails to the rear of the cab, you can approximate the bed width. Use some 1x2's and you will get a good idea what you have to work with. For fender size, mount up your running boards, the rear fenders are the same width as they are.
Joe