After upgrading my phone to 10586, my store app is still stuck on the old version and searching for updated apps does not work. How do I update the Store app?
You update the store trough the store, so if your store can't update any other apps, it won't also be able to update itself. So, I guess you're stuck with that version.
The newest one is 2015.23.23.0
I did a hard reset on my AT&T lumia 950 and it didn't solve the issue. I have even exchanged the phone and used the windows device recovery tool and nothing.