If the owner of a vehicle consents to the placement of a GPS device on her vehicle, does another person who regularly operates that vehicle have standing to complain that the installation of that device constitutes a violation of his 4th amendment right and therefore an illegal search?

I'd say no. You share property with someone else, you run the risk that they're going to give consent to the police. I think unless you have a Georgia v Randolph situation of the other person present and actively saying "no, don't do this," you're fine.