Coffee and tea contain polyphenols which cause headache in some people.
(Also in red wine). They dilate blood vessels and some are sensitive to this and develop headache.

I have met many people who get headaches from tea. Far more than complain about coffee. I have this problem
from time to time myself. I only get the headache when I consume large amounts however. I also get headaches from red wine-- I gave up all wine
years ago.

People with headache histories often develop caffeine withdrawal headaches
very quickly--sometimes within hours of drinking the caffeine (or eating it in candies)..leading to escalation of consumption of caffeinated drinks.
These folks may be latent migraneurs.

People come to me with every complaint imaginable..I have seen multitudes of things and over the years hundreds of thousands. (40yrs).
I have never seen a true coffee allergy-- but lots of intolerances.
Tea has been far more common as a headache trigger in my opinion. With the surge in interest for high dose antioxidant supplements, keep in mind that these too, contain polyphenols and can dilate cerebral blood vessels. People may not
think of that, and blame something else they had that day for any headache that arises.

People with ragweed allergies can cross over to foods. There are several fruits that will cross react in those patients. Also nut allergies can cross react with pollens from some trees.

A true food allergy should be tested for...because they can be life threatening.