Also it could be the end of a pipe from a sump pump that has been burried
over the years and the water has found its way to the surface.

I think you need to dig down upstream of where the water is exiting to see
if there is a pipe.

Wouldn't it be easier and simpler to just take a sample of the water
to a lab and see if there's anything in it that you don't want in your
vegetable garden?

Instead of getting a pump powerful enough to drive a sprinkler head,
abandon the sprinkler(s) entirely, and use drip irrigation, which uses
less water, anyway.