RESTful Web services are one way of providing interoperability between computer systems on the Internet. REST-compliant Web services allow requesting systems to access and manipulate textual representations of Web resources using a uniform and predefined set of stateless operations.
from Wikipedia.
This means, the REST API's only concern is to provide the data to work in a uniform and predefined set of operations, where those operations take the HTTP Verb that was used in consideration.
in your example, your GET route, should only be api/brands/{brandName}
the default in rest api, http verbs say:
GET - getting one element or a list
POST - creating
PUT - updating
DELETE - removing
in your application, the best approach would be something like:
GET /api/brands will get ALL existing brands
GET /api/brands/<brand_name> will get just one brand
POST /api/brands will create a new brand
PUT /api/brands will edit an existing brand
DELETE /api/brands will delete an existing brand
from your question:
Now I would like User to choose brandName from selection list on the website
the website would then request a GET to the route /api/brands to get the list of all of the brands.
This is the REST API part, it concerns ONLY in providing the right data to the system that request it.
if you want to create a website in order to CONSUME this data, you can easily create a new web project in your solution and request the data that the API provides, making the Website completely "blind" from where the data comes from, as it only asks for the data itself.
Making the whole system much easier for updated and maintainability.