What types of cables support closed captioning? For example, I know that HDMI cables explicitly do not send a CC signal to the TV, even if the source and TV both support it.

I assume that coax cable does support it since I can use a digital antenna and receive CC on my TV from digital over the air broadcasts (in the US).

Is there any other cabling that will carry the actual CC signal for the TV to decode?

I am currently sending a DVI signal to projectors in an auditorium and I am sending the same image by VGA over CAT5 to HDTVs in our lobby. We may need to add Closed Captioning to the lobby feed only so I do not want to change the image going to the projectors in the auditorium.
Your question was initially confusing because you had focused on "cables" which is not the correct way of looking at this.

Once you've decoded a video signal to components, the closed captions are no longer available. If they were not inserted as open captions (aka subtitles) then they're simply not there.

So any transport that removes the Line 21 caption data or the digital caption packets eliminates your ability to defer captioning at the distant end -- you must insert the captions / subtitles before transport. That's the case with DVI / VGA / HDMI.

If you must allow the ability to choose a captioned or non-captioned display at the distant end, you would have to send two component feeds (e.g. two CAT5s), one captioned and one plain, and A/B switch them at that end.