2003 Windstar possible alternator problem.

Van sat in the garage while we were away for three weeks. Went to start it and got nothing, no clicking, no interior lights, etc. Figured battery, which is three years old, had drained and called AAA. Guy came over, hooked up a battery back to jump it and the car started with no problem. As soon as he disconnected the battery pack, car shut down.

Repeated the exercise two more times, pretty much same result.

Is the problem with the battery, alternator (which is the original), or something else? If it's the alternator, how difficult is it to replace?

It sounds like the battery is dead from what you describe.
Take note that sometimes both the battery and alternator fail together......a bad battery can place a large load on the alternator.
Also, a weak alternator can cause your battery to discharge.
The AAA person should have been able to give you more help than what they did.

Since the battery was discharged down to nothing......you would be best off replacing it.
Many auto part stores can test your alternator for you.
If the new battery does not charge up.....then you would be looking at a new alternator.......but start with a battery and see if that solves the problem.....because you KNOW that is is completely dead......and trying to run the vehicle with it for any length of time is very likely to damage the alternator.