What year trucks will work to swap body parts?

Those parts are all shared from 2007 up to 2010 on the Sport Trac. The front fenders and doors are the same as the regular Explorers in those years as well. The rear doors are different though, they are Sport Trac only.
Looking at the picture though I would say you likely have some damage to the B pillar (that is the one between the doors) as it looks like it may be pushed in as well as the rocker panel under the door. To straighten that will take some pulling with a body dozer and possibly cutting and welding to replace it. I would probably start by having a body shop put it on a frame table and checking the measurements before buying any parts. You might be able to buy a complete side off of a salvage yard and get all the parts needed that way.

You might look on Car-Part. Com and see which parts you can find closer to you.

The doors are shared across the Sport tracs. They are different than the ones used on the explorers though. So, if you found say the left front door and front fender off a 2009 4 door Explorer they would fit, but the rear doors wouldn't. If I were doing the repair myself, I'd probably hunt up a sport trac with a bad engine or transmission of the same year then swap those parts out of the damaged truck you have. Then strip off any good parts and sell them on marketplace or similar. Might be cheaper overall than straightening that one and replacing those panels and then painting them. Plus, it looks like at least the curtain airbag deployed and those are not cheap to replace as you cannot use a used one for the repair.

A complete door with everything on it isn't light, most of them are 150 pounds or so with the trim, glass, window regulators and the rest. The normal routine is to strap them to a pallet and ship by truck. One thing you might try, find out if a local business would let you use their loading dock, shipping is usually cheaper to an industrial address than residential.
Plus, you wouldn't need a fork truck or liftgate truck to deliver which can save more.