Can I remove a wall stud to install a pet door?
If this is a load-bearing wall you'll want to frame the pet door in, just like you would a window.  This will allow you to properly carry the load down, and around the pet door.



The king studs should go all the way from the top plate, down to the bottom plate. The header will be made up of 1/2" plywood/OSB, sandwiched between two 2x4s.

If you don't want to tear out the drywall all the way to the ceiling, you could use the studs to the left and right of the stud you want to remove as king studs. 



NOTE:
Since you're only dealing with one stud, you probably don't have to worry about carrying the load while you work. If however, you're removing multiple load bearing studs, you'll want to use bracing to carry the load until you've replaced the load bearing structure.