It's not too early for job hunting, although you are shooting a little too high for your experience level.
What you should be looking for now is an internship somewhere. Many companies are looking for students to do basic coding for them over the summers. Yes, you can even do the Web development you were looking for. When I was in intern, I worked on several things: Code cleanup, a program to audit a database and report stats, experimenting with new methods and writing & documenting a sample program, etc. It can be good summer employment while you are on summer break. And when you graduate, you'll have a ton of experience for your resume.
Finding internships may be challenging. Check with your job placement office at your university. They likely have a list of places that are looking to hire paid interns. Start with that.
You might also call around to places that you like and ask to speak to a manager in the software (or web) development group. Then just ask if they are hiring interns, introduce yourself, tell them what you are interested in. You might get lucky and find someone to hire you, even if they weren't planning on hiring an intern.
Remember, even if they aren't hiring interns or don't have a slot for you, they might be able to point you to someone who does. Often, IT managers know other IT managers at other companies, so they might at least be able to recommend somewhere to call where you'd have more luck.
Focus on getting the experience through summer internships, and you'll be able to find a great job when you graduate. I did that when I was at university, and it led to a job at a company I interned for, after I graduated, and I didn't even have a CS degree (I was physics).