Microsoft has rolled out a new update to the OneDrive for Android app, bringing with it fingerprint authentication support. The new update is version 5.14 for those keeping score at home and has been released to the Google Play Store.
When you update to this new version, under the app settings, you will have the option to enable a PIN code as you have been able to do for some time in OneDrive. Once you enter that PIN, you will be prompted for your fingerprint. You can either tap your fingerprint sensor or skip this if you want to continue to use your PIN to log in and not your fingerprint.