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* (in AmE), not a walk-up order. [But see @Muzer's answer about "walk-up fares|prices", which are terms used in transportation contexts in both BrE and AmE".]

P.S. I know that "walk-in" is used in AmE, and it [seems to be used in BrE as well][2], though perhaps the term is just catching on in England, *since it is such a cultural backwater :p*

> 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][3].

> "...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).


  [1]: http://dictionary.cambridge.org/us/dictionary/english/walk-up-price
  [2]: https://www.justeatplc.com/news-and-media/press-releases/just-eat-adds-innovative-electronic-point-sale-technology-meal2go-acquisition
  [3]: https://books.google.com/books?id=z3I_wzri-SUC&pg=PA280&lpg=PA280&dq=walk-up%20flats%20in%20Britain&source=bl&ots=wZnDx3yAxK&sig=aiBXbcq7drYBFmSokC7ZuxtEAhY&hl=en&sa=X&ved=0ahUKEwjT-YGnx4jVAhVLNT4KHZiTBCsQ6AEIaDAN#v=onepage&q=walk-up%20flats%20in%20Britain&f=false