That would work as well, but it runs the risk of confusing it for "she writes on the letter", and may be perceived by some as not transitive (i.e. doesn't take an object such as brevet), whereas skriver under always means to put your signature on it.

(This answer has been edited, in case somebody saw the original one. I had a brainfart.)

Adding hon skriver på brevet. Just remember that this is a particle verb, so that på needs to be strongly stressed. If you don't stress på, it will only be a preposition, and then you'll get the meaning 'She writes on the letter' instead.
We normally only use the particle verb skriva på with nouns like kontrakt 'contract' and similar where it's clear that it means sign. For letters, the most idiomatic way to say it is skriver under.