I was going to suggest
Little Church of the West, too. It *is* almost on "The Strip," just south of Mandalay Bay on the other side of the street. But it is free standing, and has a small garden area you can use for the ceremony as well as a chapel (which seats 50). I think its tasteful; we renewed our wedding vows there in 2002!
You can see pictures of the chapel at the link above, and also find a price list for their services, which are reasonable.
If it is a marriage, and not a renewal of vows, you have to have a Clark County Marriage License.
This web page explains where to get it. You cna fill out the application beforehand and bring it with you. I am not sure, but they may only take cash for the fee ($55.) You'll find many "hawkers" there who want to hand you fliers (or even take you) for various places close by which have bargain basement prices. I can't recommend any of them, cause I'm not familiar with them.
If all you want to do is get married, and there are no friends or relatives attending (or only witnesses), you might even consider being married by one of Las Vegas' Marriage Commissioners. We have some friends who did that, and it was inexpensive, quick, and tasteful. The commissioner even allowed them time to take pictures of their hands with the rings on together on the pillow after the ceremony. This office is right around the corner from the license office.
Fern