I've heard of "SETI@home" and "Galaxy Zoo". Is there a similar program with SPACEWATCH®?

Try going to this site:  https://www.zooniverse.org/projects/fulsdavid/the-daily-minor-planet to volunteer to look for asteroids.  It is run by the Catalina Sky Survey https://catalina.lpl.arizona.edu/ at the University of Arizona's Lunar and Planetary Laboratory.

In 2003, SPACEWATCH® started the "FMO Project" courtesy of funding from the Paul G. Allen Charitable Foundation. This effort ended successfully and fruitfully in Spring of 2006. To read about the project and to see the results, you may access our FMO Project webpage through FMO Project  and https://spacewatch.lpl.arizona.edu/fmo-discoveries-and-recoveries  and  https://ui.adsabs.harvard.edu/abs/2005MPBu...32...53M/abstract .