Where is the Puzzle Gallery?
Sorry if the explanation for the Puzzle Gallery is a bit unclear.
The Puzzle Gallery is unlocked if you complete the entire Item Collection of any Deck (except for "Alone in Space").
If you have completed at least one Deck's Item Collection, you'll see a small yellow frame with a question mark in the Item Collection page of the completed Deck.
For example, if you have completed the "Classic" Deck, go to your Collection and find the small frame in the Classic Deck's Item Collection page.