I have noticed for the last week or so my toilet has been clogged off and on. I would plunge it, and it would be fine for a few days, then it would clog again. This evening I was shaving my face and I noticed a puddle forming either under or behind the vanity. I quickly shut the water off. I noticed that the water was also coming from behind the vanity in that bathroom as well (same wall sinks back to back.) If the sink water is off I do not see water. Any thoughts?

I think they are unrelated. Old pipes like cast iron can get narrow inside after many years, from hardened deposits along the walls. Or there may be an object snagged somewhere down the line, or just a clump of paper, hair, and other matter. So that's the drain problem.
I would guess that your water leak is from the underside of the faucet. Feel for wetness up as far as you can reach under the vanity, from the shutoff valves on up. Or wrap toilet paper around the spots that might leak up in there and check it after you run the water. Open hot only, check it, then cold.

If your toilet has been clogging off and on for the past few weeks you may need to have the sewer cleaned. The back up of sewage could also damage connections behind the walls to other fixtures. Be sure the floor and vanity are dry. Pour a bucket of water down the sink drain and see if the water leaks from behind the wall and around the vanity. Doing this will rule out a leak from the faucet.
If the water appears, you may need to expose the leak to determine what to do next.