Get a Programming Internship if you don't know programming? Learning instead?

If you really want to know if it is possible... yes.

If it is a good thing for you or the company is what I would really ask yourself. I see good and bad things about it.

Good: You will learn a lot about what you don't know. You will find out if you really would like to be a full-time developer. And hopefully, you will learn enough to make you want to go to school and learn some more theory.

Bad: If you get into a job where you are just doing grunt work you may find it very unrewarding and decide you don't want to be a programmer. You may get into the position and find yourself in over your head and there may be nobody to help. If you have a lot of potential employeers in your area this may not be a big deal. But if you don't and someone remembers you as "that intern that couldn't get anything done" that might come back to bite you.

Likely, a company is going to be more concerned about whether you have the skills necessary to do their intern level work than whether you have completed particular course-work in programming. So on the one hand, only one formal class in programming may be sufficient, if you have enough additional knowledge/skills to apply, however, you may need another year or two of programming courses, such as data structures and algorithms, to have enough background if this first course in programming is all you've ever done.

Most companies that I have worked for would generally only hire interns who had a minimum of two years of programming course-work under their belt. While to some degree most intern positions will allow you to learn and expand your programming skill-set on the job, it would be unusual to find a company who would pay you to learn the basics and fundamentals of programming on the job. It really depends on the job description though and whether you have enough skills to do the job they need, even if you muddle through it slowly.

I agree with Bob in the sense that you will probably end up finding an internship in some company, since I reckon the demand is always increasing for "basic" programmers which are cheap yet efficient in what they're asked to do.

However, I believe it's not a good idea to start working if you still want to learn; the company will help you become proficient in the languages you know, maybe will help you acquire the required knowledge for other technologies they use, but I do not think you will be given the opportunity to study the foundations of Computer Science, algorithmics in details, cutting-edge design patterns, functional-programming paradigm.

In a sense, you might be put in a "box" and be expected to stay there with the Visual Basic and SQL programmers -- only.

If you feel you really want to know more about Computer Sciences (beyond "coding" which is only a subset of the field), you should head for a university and get an MSc in Computer Science. Otherwise, you're fine starting your career as an intern, but I'd suggest you to keep reading books/blogs, and consulting websites such as this one to increase you knowledge.

I think that most companies are going to assume that you are a competent programmer, and knowledgeable in desired fields.

You don't need to have taken any classes (though proper schooling certainly helps your credibility), as long as you have the requisite skills to do the job (and this is true for many fields of work). If you find an internship that meets your skill level in a specific language or discipline, I'm sure it'll work out.

Realize that companies will be more forgiving of interns, and part of the deal is that they'll teach you a thing or two, and give you some practical work experience - however, the other end of the deal is that you provide them with cheap, but useful, labor.

I would say that it's unfair to assume a company is going to teach you anything and everything you need to know to perform the job correctly; on the other hand, it is fair to assume that the company will give you basic training in their system so as you can do your job. Bottom line, you need to know something before you can get a job or internship.

I would recommend that you ask around, and see if anything fits. I would also recommend that you give the Programmer Competency Matrix a try, if you do decently, you should do fine at any entry-level job/internship.