Can Apple Store install iPhone screen protectors purchased through the Store app?

Many official Apple Stores offer in-store fitting of Belkin screen protectors that are available to purchase at the store. Many official Apple Stores have a special fixture with which to accurately align and apply the iPhone screen protector's that they sell.

Whether or not the store will be able to fit a screen protector purchased via a different retail channel is unknown/unpublished. As such, if you require in-store fitting, you would be best advised to enquire at the store of your preference before purchasing from elsewhere.

You should note that Apple's online and in-store pricing for iPhone Accessories are the same. As such, given that you propose to visit the store anyway, you would be best advised to purchase the screen protector in-store. In so doing you will remove any doubt as to fitting availability.

they should be able to do it as long as you show them the proof of purchase.