I got the shop working! You can now buy a new ship, choose which ships will be a part of your squad, and equip upgrades. You only earn an upgrade if you completely clear a level of enemies though, so it might take you a few tries to get the first upgrade. The new ship also requires a bit of explaining. It seems to act like a shield ship without a shield, but what it is actually for is destroying the boss in one hit. Once you see the boss appear, drag the new ship into it, and watch what happens.