Does a chromebook need wifi to work?

Except for very simple things, yes.

The Chromebook and its operating system Chrome OS were designed to operate as clients over the internet to cloud based servers, which is where most programs for Chromebooks actually run (not native on the Chromebook itself as on other laptops). As Chromebooks access the internet using wifi, they cannot access the cloud based servers without wifi and will not be able to work for any programs that actually run on those servers.