Does Touch ID for Mac Touch Bar support third-party apps?
Yes.

Apple has published a guide - KeychainTouchID: Using Touch ID with Keychain and LocalAuthentication for integrating Keychain with Touch ID.  This means 3rd party developers can create code that allows users to authenticate to their app (an even possibly their website) via the Touch ID mechanism with credentials stored in Keychain

Additionally, more info can be found in the LocalAuthentication documentation

As for the Touch Bar... Adobe Photoshop currently support it.   

According to the macOS Human Interface Guidlines, third party developers will be able to design their apps to use the Touch bar. Links to the API documentation and sample code are included.  So, it's only a matter of time before apps get updated to support the new bar.