Yes Mussorie has some rafting..although it is offbeat activity in mussorie Coz people much prefer zip lining,rock climbing etc..but still you can opt for half-day tour and can raft from Barkot to Lakhamandal and Damta to Yamuna Bridge, near Mussoorie.
Check the website of Uttarakhand tourism board..you can book from there but in monsoon rafting is closed everywhere so check before hand for weather..HAPPY TRAVELLING
River Rafting is only possible in a place with an evergreen river, and Mussorrie is a hilly town. You can avail other adventure sports here, which doesnt requires a river. In case you are only bent to river rafting, than Rishikesh is the option. Go by Ddun to Rishikesh. It would take you maxx 2 hours by Ddun.