I haven't bought anything using an education discount in over 10 years but my impression was that these days verification is done online and presumably you have been able to do that.
As for pickup, this is what Apple says here --> https://www.apple.com/legal/sales-support/sales-policies/online-store-education-individuals.html
In-Store Pickup and Return
Apple offers in-store pickup for
many of the items available on the Online Store. Certain products and
payment methods, however, may not qualify for in-store pickup. Only you
or the person designated by you may pick up the item(s) purchased. A
government-issued photo ID and order number will be required for pickup.
Apple will notify you when your order is ready and the date by which
you need to pick up your items. We’ll also send you reminder or two,
just in case it slips your mind. If you don’t pick up your order, Apple
may cancel it. Please visit http://apple.com/us/shop/browse/home/personal_pickup for complete details about in-store pickup.
You don't say in which country you are located so you may want to check the version for your country.