The word I'm looking for means to capitalize the first letter of some words, but I want to use a single verb since I need to say this many times, as in "*** all the words that appear in the table headers".
Is there such a verb?
When you speak of capitalizing a word, that normally means to capitalize only the first letter. If you were to say "capitalize all the words that appear in the table headers", I believe this would get your meaning across fine (and as always in specifications, a little example goes a very long way).
The only other likely interpretation, putting it in all caps, is not normally expressed that way; usually one would have to say "put the words that appear in the table headers in all caps" to communicate that.
In the programming community, this activity (capitalizing the first letter of every word) is known as "Propercasing" ...
Title Case (Some Short Words Don't Qualify for Capitalization)
The name "Proper Case" comes from the capitalization rules for proper nouns. See Wikipedia's article on Sentence Case.
I agree with chaos, but you could also say "put the table headers in title case".
Unfortunately that doesn't completely remove the ambiguity, since styles vary on whether prepositions, conjunctions and so on should be capitalised. The page linked also mentions start case as having the first letter of every word capitalized, but I must admit I've never seen that term (start case) used before.
Besides capitalize, my first instinct on the matter, "put into uppercase" could be another option.