I've noticed quite a number of religious professionals of late have used phrases such as "let's exegete this text" or "we need to exegete Paul's meaning here." Of course, an exegete is one skilled in exegesis, but I have never heard of "to exegete" being in accepted usage. Wiktionary included it as a possibility, but I would hardly consider that an authoritative source. Roger Wibberly critiqued a book titled Medieval Music as Medieval Exegesis in Music and Letters 82.2 (2001): 291-293 and argued that "the often-used verb 'to exegete' strikes a note of literary discord within a generally elegant text."
Is the verb "to exegete" becoming accepted usage? Obviously, it has been used consistently in at least one published source and is currently being widely used in religious circles.