Will Guitar Center switch out pickups for me?

If not, where would be a good place to look?

They do, but it'd be better if you learned how to do it yourself. Get a soldering kit from RadioShack or something and maybe look up a few videos on YouTube or ask someone if they can teach you how to solder.
Personally, I'd look for a local, privately owned shop to do it before I'd take it to GC. I'd say you'd have a better chance of it being done right since the shop owner is held accountable, as opposed to whatever tech GC has on staff.
My GC has a luthier that works everyday. He said he'd do 2 humbuckers for me for 45 bucks.

Hohner MR-800
B.C. Rich Rich Bich Platinum
MIC Squier Strat
Boss MT-2 Metalzone
Fab chorus
Fab echo
Fender Super champ XD
yes thats fine
but pick ups arent really hard to change yourself if you find a diagram on the net...but if your not confident in your mad skillz at changing pickups...go to guitar center...or maybe one of your friends could do it for free? idk...ive got some hommies that can do anything...i mean ANYTHING from fixing cars, to cutting hair, to giving tatoos, to cooking, anything...so see if you've got that first...
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Better than hooked on crack, I suppose. I'd rather know my kids are safe at home beating their meat than out in the world robbing old women for their crack fix.

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What a huge coincidence. I have a butthole also.