The GitBucket api/webhook is designed for compatibility with GitHub's implementation.
The API endpoints are reachable under: http(s)://yourgitbucket/api/v3
GitBucket does not fully implement the GitHub API/webhook and respects only a subset of parameters. If you find some behavior different from GitHub, it can be a GitBucket bug or a feature not yet implemented :).
With a default local installation, having created a token for the root user, calling
curl -H "Authorization: token 6b690aa9e528c54835619b2cb717f61035e9a013" http://localhost:8080/api/v3/user
answers
{"login":"root","email":"root@localhost","type":"User","site_admin":true,"created_at":"2016-04-01T13:27:12Z","url":"http://localhost:8080/api/v3/users/root","html_url":"http://localhost:8080/root"}