A walk-up is an apartment in a building that lacks an elevator.
A walk-in is a person who comes into an establishment without an appointment or without having phoned beforehand. A walk-in order is an order placed by such a person. Many different kinds of establishments refer to "walk-ins" to describe some of their customers: health clinics, car dealerships, restaurants, spas and salons, and so forth.
You can walk up to the counter and place an order. But such an order would be called a counter order, not a walk-up order.
P.S. I know that "walk-in" is used in AmE, and it seems to be used in BrE as well, though perhaps the term is just catching on in England:
With this tailor-made EPOS technology, JUST EAT will now be able to offer its takeaway restaurant partners a central system for managing orders, whether online, ‘walk-in’ or over the phone.
And "walk-up" is used in England as well.
"...which tended to be composed entirely ... of walk-up tenement flats..." (Cambridge Cultural History of Britain: Volume 9, Modern Britain edited by Boris Ford. 1992).