I'm creating a website that uses only the Google book search API, should I use the api key every time the user sends a request searching for a specific book? And is there a limit if users send requests without my api key?
According to Google Books API Docs you can perform a search without an API key, so technically there is no limit. Also you don't have to use an API key on every search. However, if the call is made with API key, each Volume will include user-specific information, such as purchased status.
Note: Performing a search does not require authentication, so you do not have to provide the Authorization HTTP header with the GET request. However, if the call is made with authentication, each Volume will include user-specific information, such as purchased status.