People who write are "writers," though many call themselves "authors," especially if their products are books, or legislation. More and more, they say that they "authored" what they wrote.
"Author" as a verb is what is politely called "disputed usage." (It’s also called, less politely, "self-centered balderdash" and "pomposity.")
Garner’s Modern American Usage notes that, while "author" is close to being standard, it is "a highfalutin substitute for write, compose, or create," and is only at Stage 3 of its five-stage Language-Change Index, meaning that just because people use it, it don’t make it right. Garner’s particularly abhors "author" as a verb applied to lawmakers sponsoring a bill: "This seems irresponsible, given that few legislators today actually write the bills they promote." (Garner’s curiously lists the verb "co-author" as standard.)
A usage note in The American Heritage Dictionary says "author" as a verb is often criticized as a "pretentious synonym of write," and should never be used for unpublished works, such as love letters. And while its usage panel still sides with those who oppose the verb "author," "this sympathy has been slowly eroding over the decades." This dictionary, however, is "more tolerant" of the legislative use of "author." Most important, it predicts that "the verb will eventually be accepted by most people."
But the Oxford English Dictionary says "author" has been used as a verb since 1598, even as it lists it as obsolete in US usage. So why it is not completely standard is something of a mystery. Anyone want to author that?