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!