As an undergrad student, how should I ask a professor for an opportunity to do research with them?

Absolutely, with no qualifications, approach the researcher and ask what it would take to do research in his/her lab. As a first-year undergraduate student, it's very likely that you don't have the knowledge to contribute to the projects. So approach humbly, asking what you can do to prepare yourself. Ask for suggestions for a review paper or a few research reports that would bring you up to speed. Ask if you can attend lab meetings as a fly-on-the-wall to absorb the controversies and decisions being made day-by-day. The very worst possible case is that you're told, "no." If so, you're already there. Next worst is that you invest your time and get only an education out of it. That's not so bad. You'll be positioned very very well for next year.

Good luck!

Personally, I am always happy to talk about possibilities for working in my research group with a student who's interested in the group. The challenge is that most first-year students might not have the background to understand what's going on in the research, so they end up being more "technicians" than researchers.

So I don't think it's a problem for you to discuss the possibility of doing the "internship"; your university may even offer an opportunity to get course credit for it as an "independent study." However, you should be prepared for the possibility that the professor feels you're not quite ready for the work—but he may definitely suggest ways to overcome that deficit.

Here's another option: email or talk to one of the professor's graduate students. By working with the graduate student, you'll be working with the professor anyway.

For context: I'm a computer science PhD student at a large United States research university. At my school, most professors are bombarded with emails and requests from undergraduates (in addition to, of course, PhD and masters students, and postdocs). Indeed, my PhD advisor told me that he/she gets about one email each day from an undergrad requesting research opportunities.

I've worked with roughly 20 undergraduate researchers to some extent. About half of them got in our lab by emailing PhD students, who then recommended him/her to the professor if the undergraduate's resume exceeded a certain bar. The other half emailed the professor directly, but often in those cases, the professor ended up forwarding the undergrad's resume to the PhD students. While I don't have hard statistics on what this is like throughout my entire department (let alone university) I think this is a common strategy that students pursue, and the one I would begin with if I were back in my undergraduate days.

As another positive note, sometimes graduate students' research websites are more up to date than those of the professor's!