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"O" or "R1" as usual. If you do not have the swingshot then you've encountered a glitch.

The little pad thing on the left of the controller is another inventory quick select. You use the farthest one to the left which equips the swingshot, or you can just jump to mid-air and hit whichever button you use to fire weapons or launch bombs, in that case, it automatically equips it, and you still have your other weapon to use when you are finished