Diplomas can help get you past the initial HR screening, but most places will hire anyone who shows they have the desired skills and attitude. Where I work now, maybe 1/4 to 1/3 of us developers has a relevant technical degree, the rest of us (including me) have unrelated degrees (I have a bachelor of fine arts in music education). If you don't have a relevant degree, though, then networking (in the people sense of the term) becomes more important, at least to get the first job or two onto your resume.
I'm not saying don't go to school if the opportunity is there, but not to worry as much about a degree as picking the classes that will teach you the fundamentals (general programming theory and techniques if going the software developer route, or color theory, text layout, etc. if going the web designer route) so that you get a good grasp of the fundamentals -- while all along spending time on your own learning web-specific skills and tools, speeding up the process to where you're ready to do the work. In the mean time, meet with local users groups and tech meetups, network with like-minded people on the web, join open source projects where you feel you can contribute, etc., in order to be on people's minds when they have an opening at their company -- an employee referral can be a lot more effective than any degrees listed on your resume (unless maybe you're applying at NASA or such ).