The internet was disappointing. Some things stated that it could possibly be genetic and some just provided "some people can, some people can't," which is not very satisfying. I looked up various muscles in the hand for snapping your fingers, but that was disappointing as well.
It seems that why some people can't snap their fingers is one of the great mysteries of the world. I did find out what makes the sound though, which I had never thought about before. Ima just straight copy the wiki entry for that because...a lot of steps.
"There are three components to the snapping finger sound: (1) The "friction" or "sliding" sound between the second finger and the thumb (2) The "impact" sound from the second finger colliding with the palm and (3) The "pop" sound from the rapid compression and subsequent decompression of air. The third "pop" sound is the most audible of the three components and is caused by a compression of air between the second finger and the groove created between the palm and the third (ring) or fourth (little) finger. These last two fingers aid in resonating the "pop" sound within the hand. If the third or fourth finger is not in contact with the palm, only the first two components will be heard and there will be a significant reduction in the total "snap" sound."
I did find an alternative way of snapping for people who are incapable of regular snapping. Essentially you touch your thumb to your middle finger and fling your entire arm, which makes your loose index finger snap against your middle finger that is touching your thumb. Really.