I put a new lockable gas cap on my 2004 Jeep Wrangler. My check engine
light is now on. Could the new gas cap have caused it, and if so, how
do I turn off the light?

you might wanna put your old cap back on, and see if the light comes
on, if it doesnt, then it just might not like the new cap you put on,
newer models do have sensors on the gas caps

Grrrr to gas cap sensors!  Indeed it is likely the reason your check
engine light is on (since you just put the new one on), I agree with
Neo that you check the old one to see if that is the problem.

But to go on a rant, I'm tired of mechanics telling me I didn't
tighten the gas cap enough every time I take my check engine light in
to get checked.  They tell me this before they check anything even if
I tell them a problem I'm having with the car that I think is related
with the light.  It's so obnoxious!

OK, I'm done ranting now and going back to being plain ol' Jack again.

The check engine light indicates that there is a stored trouble code
in the onboard computer (at least one actually).
Your local auto parts store should have a reader for a couple of bucks
that will tell you exactly what's wrong.