OK, so it turns out this is really easy. In Audacity, click on the Generate menu entry and select Rhythm Track.
In the Rhythm Track window, set all of the parameters you want. Use the Number of Measures entry to make the track as long as you need. I recommend over estimating the length you need since you’ll end up deleting the click track anyway.
If you plan on having a bass in the mix, I recommend recording it first. The bass is what our ears tune in to for a rhythm in the absence of a hard beat like drums. I’ve always had the best success with multitrack recording when I record bass first, and I think Ben has said the same thing.
I may have to try Audacity for my next project. Let me know if you need anything else.