Just be aware that working as an independent, free-lance web developer -- and actually making any kind of decent money at it -- means you not only have to be good (and fast) at the technical stuff (and also artistic stuff if you plan to do the whole shebang), but you also have to wear several other hats: sales, requirements analyst, cost estimation, accounts receivable, customer account manager, QA/testing, etc.; as well as making sure you have crossed and dotted all the legal T's and I's. All the while, you'll be competing against someone in a 3rd-world sweat shop or some 14-year-old willing to work for a few bucks an hour to do the same thing.
Can it be done: I'm sure it can, but not easily.
If, on the other hand, all you're really interested in doing is the actual technical stuff, life is probably easier working for a dot.com or other company in the business where you can concentrate on doing what you do well, while other people do what they do well (sales, accounting, etc.). Yes, this may mean you can't make your own hours and work from wherever you want, but many tech companies are pretty flexible with schedules, working off-site, etc. Having a degree in an applicable field can help you get your foot in the door at such places, at least past the first resume screenings (though personal networking is probably more effective: an internal recommendation can be huge).
Don't mean to be sounding like you should forget your dreams: just go into it with your eyes open. If you still think that's what you really want to do, by all means go for it -- especially if you don't have to be responsible for supporting a child or other such responsibilities right now, in which case there's no time like the present. You could still take a class or two per semester and continue toward your degree, perhaps? For that matter, you could just continue with your present schooling, perhaps taking close to minimal hours, and spend your free time developing web sites for pay instead of spending money on beer.