If I understand the documentation correctly, it should be possible to retrieve user Steam library data using their API key even when their library is private. That may, however, violate the Terms of Use 
- You agree to keep your Steam Web API key confidential, and not to share it with any third party. This license is personal to you and specific to your Application. You agree that you will be personally responsible for the use of your Steam Web API key.
I’m not saying this is worth the effort, but one solution would be to create a browser plugin that uses the key to download Steam library data, and upload it to Grouvee. This way, the key would never leave the user’s PC.
A browser plugin could also be extended to retrieving and parsing library data from other services, cf. Request: Import games from other stores besides Steam