I'll say the following based on my experience (I started my career with only a AA degree, and later I earned by BS and went to grad school).
"I've got about a year of computer science classes under my belt along with countless hours of independent online and tech book learning. I can put together a secure login-driven Web site using PHP and MySQL. (I have a personal project on GitHub and a personal Web site.) I really enjoyed my Web development class, so I've spent a lot of time honing those skills and trying to learn new technologies. I still have a ways to go, though. I've been designing Web sites for more than 10 years, writing basic PHP forms for about 5 or 6 years and only gotten seriously into PHP/MySQL the last 1 or 2 years on and off. I'm fluent with HTML and CSS, but I really like back-end development.
If you are doing a CS degree, you need to clearly put in your head that there is more than web development in CS. If you really want to do back-end development, you need to disabuse yourself of the notions that come with basic front-end web design. I'm not trying to be an ass, but everything that you mentioned is assumed to be very fundamental knowledge. That alone won't get you to open most doors.
I was hoping I might be able to get a job as a junior Web developer, but even those require 2+ years of experience and a list of technologies as long as my arm.
That is the sad states of technology nowadays. As I mentioned before, I started my career with a AA degree in the early 90's. That was the time when companies starting tightening the requisites. I saw the writings on the wall, and I kept studying until I got my 4-year degree in CS, and then went to grad school.
I honestly believe 70% of work done in IT (be it sysadmin or development) does not require a full-blown degree in Computer Science. Sadly, nowadays, there is no fucking way anyone can get a decent IT or development job without a 4-year degree. It is what it is, and we cannot wish it away.
Internships usually require students to be in their junior or senior year
It has been like that even during the times when a AA degree would get you an entry level programming position.
, so that doesn't seem to be an option for me. Recruiters are responding to my resume on various sites, but it's always for someone more experienced. Should I forget about trying to find a junior Web developer position after only one year of computer science classes?"
Yes, because only one year of CS classes just won't cut it. It didn't cut it before, it won't cut it now. Consider that there are a lot of people with degrees already that are looking for a job. That will be your competition, so you know that the odds are against you... for now.
What I did then, and what you can do now is to get a part-time job at your college in the computer lab, be it by tutoring, teaching or just regular IT maintenance. If you can get close to the sysadmins at your CS department, pester them until you get a job with them. There is always lots of programming opportunities in terms of sysadmin automation.
If you are lucky, they'll be working with a bug tracking system, and that exposure will put you ahead of many people upon graduation. When you work part-time for the CS department, you begin to meet people. Those connections have the potential to open doors in terms of internships when you get to your junior year.
Don't look at your CS department as just a source of courses to take, but a venue where to make professional connections. That is what I did, and it has paid itself a million times over the length of my professional life.