So apparently my van is fine. I was told yesterday by some jerk at Auto Select that I needed a new alternator, and that it would cost me $300. Since those guys wouldn't install an inexpensive used part, this morning I took the Deathtrap to what is possibly the best auto service center in Appleton.
And now it appears that — to use the mechanic's delightful greasemonkey colloquialism — the guy at Auto Select was just "pulling my dick." My alternator is fine, and my battery (which just needed a good jump after all) is good enough for now, though I should probably replace it at some point.
The mechanic at Christy's Service didn't even recommend I get it replaced at his garage, noting that Auto Zone installs batteries for free.
Total cost to find out my car doesn't yet need to be fixed: $2.