I'm trying to figure out if we can take the transmission out of our 2002 mercury mountaineer which is rear wheel drive and put it in our 2005 mercury mountaineer which is all wheel drive they both have the same 4.0liter engine.

 

Physically, maybe !

CORRECT functioning, most likely NO.
The electronics controlling the 2005 may not be available in the 2002. Too many electronic components and wiring to do the job.

Choices would probably be, get the 2002 trans repaired and reinstall or hope to get a good trans from a salvage yard.
What is the original problem, maybe can be corrected without too much trouble?

 

The 2002 (Rwd)mercury mountaineers transmission was just rebuilt last year the 2005 (Awd) mountaineer I have no idea what the problem with the transmission is but it's alot nicer than the 2002 so long story short my best friends moms mountaineer (2002) needs the motor rebuilt and it's her only vehicle and it would be so much easier to just swap transmissions and then rebuild the motor as I have time honestly not to worried about the 2005(Awd) being Awd if it will work without Awd that's fine

 

If $$ permits, What if,,,,,,,,,,,,,,,,,
arrangements were made to have the rebuild done and get a time required to do the job.

Check what a rental would cost for that period of time, just maybe, it might be the cheapest option in the long run, with the least amount of inconvenience.

Sometimes another point of view helps.

I always question if whoever is going to do the "Rebuild" has the experience and tools, (micrometers, hone equip etc.) to do a real rebuild, otherwise it is just a waste of time and money.