Generally speaking, no. Broken appliances are only going to be a headache for the donation center as they then have to figure out how to get rid of them. However, it’s worth calling and asking. Some centers have staff on hand who know how to repair appliances or may have someone willing to put the work in so that a family in need can benefit.
If donation is completely off the table, remember that dumpster rental companies will pick up old appliances. Junk removal companies, area recyclers and your local bulk curbside collection are options as well.