Possible origin is the Instrument Landing System.
When flown correctly the vertical and horizontal guidance indicators are
centered in the face of the display. These crossed indicators have the
appearance of the crosshairs of a telescopic sight.
Hence, you shoot the approach. Keeping the crosshairs centered generally
assures that you will hit what you are aiming for... the runway.

Hmmm...I always thought that when you guide an object in some way down a
narrow corridor of space to a specific place to attain a goal or objective
that that was "shooting". i.e. shooting pool, shooting some hoops, shoot a
goal in hockey. Now, I just wonder why they don't call it "Shooting a
strike" (in bowling).

Kobra