Is there a text-caret Unicode character?

Well I think it’s safe to say that no, there is no such character. Part of the proposal process for requesting a Unicode glyph is to perform a search, which I did, but it came up empty.

In addition, every other search I made came up empty, including one with ShapeCatcher which found several decent stand-ins (below), but none that works quite right (is full-height).

I guess until they add one, we’ll have to use CSS to style and size one of these to fit the part.

ⵊꕯᏆ

There is a Unicode character for caret, and it's called "caret". It is U+2038: http://www.fileformat.info/info/unicode/char/2038/index.htm

You can find it in the General Punctuation category in some character mapping accessories (such as Gnome Character Map). It is not widely available, but it can be found in at least in 50 fonts that are listed on http://www.fileformat.info/info/unicode/char/2038/fontsupport.htm. You might not be happy with the design of the character in these fonts, but that's an entirely different issue.