As I understand it, it's legal under FCC rules for anyone, licensed or not, to use any frequency or mode of radio communication as necessary in an emergency. This includes, but is not limited to, police/fire/EMS radios, CB, ham (even tuned outside legal ham bands, if the hardware has the capability), using voice in digital-only or CW-only sub-bands, etc.
That said, a key to emergency preparedness is practice -- and since your practice isn't a real emergency, anyone transmitting during a practice scenario must be properly licensed and using legal equipment and permitted bands.
In the case you describe, I don't see that your goal is different from S&R groups, run/bicycle road race groups, and others who routinely either enlist local hams for their event communications (unquestionably legal, as long as the event isn't a business operation), or go through the process of licensing their own members.
Technician licenses are surprisingly easy and cheap to earn, and the bands that permit Technician licensees to use voice have convenient repeaters available to ensure adequate communication range. As long as the communications aren't being used for the purposes of a for-profit business (no dispatching pizza deliveries or courier services, for instance), there should be no legal problems with licensed operators and certified equipment (which is also not very expensive; a 50 W dual-band mobile/base rig and antenna installation can run under $300 per station).
Where you may run into a snag is the provision in Title 47, §97.113 about "Communications, on a regular basis, which could reasonably be furnished alternatively through other radio services." This could be interpreted to mean that since your organization could obtain a business licensed communication setup (private/shared repeater and licensed-by-group station radios, like a taxi company dispatch system), that you are required to do so for actual business communication.
The ambiguity here is that you're proposing to use these radios only during communication breakdowns (phone and cell service out, for instance). I'd recommend starting by checking with your ARRL local rep (the local ham club can most likely put you in touch).