Does such a thing exist?
Our HDTV has one set of HD component cables, and one set of DVI connection.
We have the component set up with the DVD player.
It wasn't going to be an issue, because we weren't going to get into HD OTA for a while; but we moved, can't get satellite, so are going with cable. They offer an HD box for the same cost as the digital box, so why not? I've browsed around, and it looks like the box they use has only component connection [I'll check though, the maker does offer one with DVI].

So if that's the case, I need either a component to DVI converter, or a component input switchbox of some sort. Any suggestions/recommendations on either of these would be appreciated.
Thanks.

Gotta be an electronic device. DVI is digital, component is analog. Unless your set can use the analog signal from a DVI-I connection (where both the digital and analog are carried).

http://www.keohi.com/keohihdtv/interfaces/switches.html

Considering hdtv-compatible dvi switches are very expensive and are just coming to market, I doubt a dvi-to-component adaptor exists.

Your best bet would be to get a component switch box. Your other option is to get a receiver can switch component video.