What currency does the Corner Shop use?

You can stock up at Kureo's counter without spending anything, because Vauxilo coins come free through play. Spending money on them is optional. The coins are the same ones whether you earned or bought them, so you can mix both to buy.

Vauxilo is its own currency. Lumin Crystals are the paid premium currency, sold only for real money and spent in the cash shop. Glimmer is the free currency earned through exploration and daily activities and spent in the Glimmer tab of the premium shop. If you are planning a Corner Shop purchase, it is your Vauxilo balance you need to check, not either of those.