Can I become a domain name or URL registrar?

I want to sell domain names, but not the names I own. Rather, I want to sell the available domains in the general pool of available domains such as those sold by Domain.com an GoDaddy for under $10.

Is this possible?

If not, is it feasible that I would be able to create a proxy site through which customers would be able to purchase domains through an established registrar, and I would be able to keep a cut of that purchase?

Yes. Easiest to be a reseller of somebody else. See http://www.resellerspanel.com/ for example.

GoDaddy sells domains names as a loss leader. That means it costs them more to register the domain, than they get back. The idea is to get you into the door to make money off of you using some of their other services. You could set up a business setting up domains for people as their agent where you would charge them but last I knew Network Solutions charged $25 for a name for 2 years. Though the market is pretty saturated. So it will be hard to compete with the Godaddys of the world willing to lose money to get you in the door.