How do I add a column to a Word table that contains text that will
repeat within the column's cell for every new row I add?

If it's a short text, you can also apply it as "numbering" to the whole
column. Any new row will repeat it.
In "Bullets and Numbering Numbered Customize", delete the grey number
in "Number format", type in your text (up to 30 characters, it seems), and
set the "Font" formatting to your liking.

Regards,
Klaus