Seeking Church Shops

Thats a tough one. The basilica does have a small shop in it.

This may be more tradition and not as faith-based as you're looking for, but the Market Hall had hand painted Easter eggs, some of which were painted with religious symbols. If you're going around Easter, perhaps there may be more along those lines.

https://www.google.com/search?q=easter+souvenirs+budapest&tbm=isch&source=iu&ictx=1&fir=bdL1iYlbSfZZ9M%253A%252C3by8tq7odFy6vM%252C_&vet=1&usg=AI4_-kQXwRKzFN7UckEYRsn2YQyOFIOg8g&sa=X&ved=2ahUKEwiL6_CvspfhAhUFS60KHcEkDCUQ9QEwBHoECAoQDA#imgrc=bdL1iYlbSfZZ9M:

Painted eggs are part of a prominent Easter tradition in Hungary. Within the Christian tradition eggs were painted red (to symbolize Jesus’ blood), but in recent centuries this has changed to eggs being dyed many varied colors.

Those are helpful, thank you Both!