Is there a word for where an arrow starts?

The ending point of the arrow is the tail just as the starting point is the head. See my comment for three sources using it in this manner.

Tail

tail 1 (tāl) n. 1. The posterior part of an animal, especially when elongated and extending beyond the trunk or main part of the body. 2. The bottom, rear, or hindmost part, especially: a. The lowest part of a garment such as a shirt or coat. b. The rear end of an automobile or other vehicle. c. The rear portion of the fuselage of an aircraft or the assembly of stabilizing planes and control surfaces in this portion. d. The vaned rear portion of a bomb or missile.