What is a vamp on a shoe?

Vamp: The section of upper that covers the front of the foot as far as the back as the join to the quarter.

What is the toe area of a shoe called?

Toe box
Toe box: The area at the front of the shoe that houses the toes, which should be high and wide so your toes aren’t cramped. Toe wrap: A sturdy panel of rubber or leather around the sole, extending from the outersole over the midsole onto the upper.