Is it acceptable to sign a cappella (without musical accompaniment) at a community musical theater audition?

No, this is usually not considered acceptable. Auditioners for local, community musical theater are usually expected to provide sheet music -- clearly marked with their specific cut of the material -- for the audition accompanist. This may be a physical, printed copy of the music, or an electronic version provided on a tablet computer such as an iPad. In some cases, the company holding the auditions may also allow you to play recorded accompaniment at the audition. However, this should also be avoided unless the audition instructions specifically state that this is acceptable.