Are the Missions available only on the Desktop App?
At this particular moment, yes.
Musixmatch team is working hard on bringing the Missions on mobile in the smartest way, in order to develop the best experience that Curators could ever imagine.
For now, though, you can only fulfill the Missions on the Desktop App (from version 3.6) on Windows 10 and macOS.
Download the Desktop App now, your Mission is waiting for you!