For example, I have a web site with authentication. Is there any fingerprint API, that can be used from JavaScript, to login users? Mobile OSs like Android and iOS has such API. So at least mobile version of browsers, in theory, can provide such functionality. But I can't find any information on the web about it.
Chrome 70 Beta, announced September 13, 2018, introduced web authentication.
Also enabled by default are macOS's TouchID and Android's fingerprint sensor via Web Authentication. These allow developers to access biometric authenticators through the Credential Management API's PublicKeyCredential type.