As a GS leader, I feel for you, though I've never been in this exact situation. I would say that it's not fair to insist that the girls whose moms can volunteer will get in -- that is unfair to girls whose moms work or have other commitments that mean they just can't volunteer for GS.
ADDED based on SWH and other posts: I would tread very carefully if you pursue the idea of absolutely REQUIRING parents to volunteer as a condition of their girls being accepted. Once girls are accepted you can surely then tell parents, "We expect all parents to do one task each year, such as to prepare parts for a craft etc." The expectation, and some peer pressure from other parents, goes a long way toward making that happen, I find. But making it a condition of acceptance doesn't seem like the GS way to me -- there will always be girls who through no fault of their own have parents who just cannot or will not help out, and it's penalizing those girls to make their parents' participation a condition of the girls' being part of the group. I'm a leader too so I know it's a pain if parents want the world and won't help out -- but should that be the basis for excluding a child? If a girl is accepted because a parent says "I'll do X" and then the parent doesn't do it, either because she's a slacker or because something really does happen -- would you then tell that girl she must leave the troop because M. did not do X? That would be the logical and "fair" next step if parent volunteering is made an absolute condition of girl membership. I would really talk to a SU or Council person about this if I were you (see below). This is why I would really advocate a lottery or random draw from among girls whose registrations were done by a specific date.
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I would set a date for all paperwork to be in, and then do a random pick of seven girls out of those who have their paperwork turned in at that time. It seems like the only fair way. I'm not sure why you're worried that requiring completed paperwork somehow penalizes girls who have working moms -- working moms can get the paperwork filled out by a deadline the same as those who stay at home.
BUT -- before you do anything-- Have you contacted your GS Service Unit director, or your local GS council, for help and advice here? Your Service Unit may have other leaders who have been in this same position who can advise you better than any of us can. And your SU or Council may actually have some guidelines on this situation for you.
Most of all, your SU and Council should be actively working with the former leaders of the other troop to place all 19 of these girls if they all want to continue in Scouting. It is very sad if these leaders quit and did nothing to help their girls move into other troops; the SU and Council can't help unless these departing leaders contacted them. There surely are troops out there who could take a few girls here, a few girls there -- there is NO reason why yours is the only troop in the entire area these girls can get into! If the issue is "we all go to the same school and we want a troop AT school" -- remember, "school-based troops" are just a convenience and not mandatory, despite misperceptions by many parents and girls that GS is school-based. Girls do not have to be in a troop that is composed of schoolmates. The parents should be made aware of that, and also of the fact that the troop leaders (you) do have the right in GS to decide on the size of their own troops, based on what the leaders feel they can handle.
There may be other leaders in your SU who very much want some new girls but could not take on 12 at once.
If you take seven, that leaves a dozen girls -- that is enough for an entirely new troop! Talk now to your council and SU about possibly finding a leader for them and about how to absorb the seven you feel you can take. You do not have to handle this situation alone!