Searching in Brandfolder is incredibly powerful and works the same via the API as it does in the Brandfolder UI. The only distinction between the two is some formatting requirements. Including a search parameter in an API call will require a properly formatted URL. You can find the parameters available to search on in this Knowledge Base article. You can type them as you would in the UI and use a URL formatter like this one to get the desired format.
To search within a Brandfolder, send a request to:
GET https://brandfolder.com/api/v4/brandfolders/<brandfolder_key>/assets?search=<search_query>
GET https://brandfolder.com/api/v4/sections/<section_key>/assets?search=<search_query>
GET https://brandfolder.com/api/v4/collections/<collection_key>/assets?search=<search_query>
?search=label:"<label name"
Note: you can also append additional search parameters here, assuming they have the proper url formatting.