While the other answers are good, they are extremely on the pro-degree side, and I think it's only fair to shine the light on the other side of the world.
Is finishing my CS degree critical to being employable as a web
developer if I already have good experience and an internship?
First of all, single internship as nice as it is, is not equivalent to actual paying on-the-job experience. Expectations of an intern are very low, even when comparing to those of a junior/entry level developer so getting over that barrier is not exactly a proof of being amazing and a Rockstar. Granted that most interns I've worked with do not make it that high, but still, the expectations are really low most of the time, so don't take this one positive experience and praise from every direction as more than it is.
So how to tell if you were really really exceptional as an intern, going above, beyond and possibly into the level of actual-work standard? Very easily - by getting a job offer from the company where you did your internship. It may be belted until you finish your school, but that is what companies I worked with did with the best of the best interns in the past, as that's a great way to source future employees.
Would most companies auto-reject my as an applicant for not having a
degree even if I have long-running experience with high-demand skills
in web development?
An important thing to point out is what's really in demand are people, not skills. And unless its early 2000s and you have practical experience with COBOL (which would get you hired for a big bag of money without interview), then people will be looking to hire a specific personality with given skillset. Because ultimately this is what hiring employees is about - to get people with the right mindset, attitude and skills into the workforce. It is almost never about just ticking the currently-hot skills.
With that in mind if you are the right person, you will not get rejected. The problem is how will you demonstrate it?
Degree certainly is one way, you will spend 2-4 years learning about all sort of IT-related topics. And while most of them you will never use in the real world, it at least shows that you can stick up with IT-related responsibility for that long and maintain X grades level. It certainly shows commitment and that you at least should be able to hold a conversation about theory of computer science and maybe whiteboard a simple compression algorithm, but also from a degree no one will expect you to know how to properly design, and work with, say REST API, so solving actual practical problems with any technology will require a lot more effort from senior staff.
Another way to show that you are the right person is real world experience. If you were to take the 2-4 years and commit them into commercial and opensource work, truly have full-time+ repository worth of them across the 2-4 years, then that is something quite outstanding and will do the opposite of getting you auto-rejected for an interview. Of course the problem is to have the drive, the passion and motivation to actually do all this work for 2-4 years, where unless you will manage to land an entry level job (which is tough with just the intership alone) then you will be on your own to organize all aspects of your self-driven education; And that is a tough gig. But if you will manage it, you now have 2-4 years of actual experience that, at least partially, is available online for everyone to see and check out, showing not only that you can actually create software, but that you are self-disciplined enough to do it on your own, without a paycheck driving you forward. Those personality traits are almost always of extremely high value from employers perspective, as managing, overseeing and hand-holding new employees is expensive.
I don't think that either of those choices is universally right, it's more of a matter what better fits your personality.
I know a lot of people (like myself) who would not be able to pass colleague due to simple boredom and lack of practicality - I am one of those people who can only learn by doing real-life affecting work. But then I also know people who tried for the self-driven route and wound up wasting 2 years of their life with nothing to show for it besides a crippling depression.
The safest bet probably would be to mix both of those worlds, working on the degree during the day, and at night and weekend driving open source contributions and hunting for a part time job/contracts to gain the invaluable real life experience.