RACES is based on the criteria that the station is serving a government function/need at a government site, not a home of one of the volunteers. In such cases it should be obvious that a member-owned station is not relevant, except possibly as a source of back-up to the EOC if its equipment fails. An exception is where there is no local EOC and the jurisdiction authorizes a station as an EOC alternate until such time as the jurisdiction can activate its own EOC.
Of course there are certain types of field operations that cannot be accomplished without member-owned radio equipment. Examples include emergency portable packet, fly-away digipeaters, portable repeaters, mobile and hand-held units of all kinds. Few jurisdictions have the funding for such a breadth of equipment so must rely on the people in the RACES unit, the ARES, the Civil Air Patrol, and possibly the MARS for support and help in these needs out and away from the government communications center. In other instances local government has yet to acquire equipment so must rely upon temporary or permanent volunteered or donated equipment. But, as you can see, a volunteer can be quite active without owning any personal communications equipment.