Probably. The government has never turned down a big budget government entitlement in my life time, I don't expect them to turn up their nose at the chance for yet another subsidy.
As for the inevitable objection that vouchers would result in the takeover of private education, have no fear. In the worst case scenario, the situation would revert to the status quo ante. Parents who objected to intrusive gov't regulations would simply found non-voucher-redeeming private schools, and we'd be back where we started. But that's the worst case scenario. There is every reason to believe that both conservatives AND liberals would fight tooth and nail to prevent the gov't takeover of schools that they have a strong vested-interest in. Right now, liberals and conservatives fight against EACH OTHER for the control of their children's schools. If both groups had their own schools they would each fight THE GOV'T instead.
Secondly, once private schools begin to rely more and more on government vouchers for their income, they'll have to acquiesce to the demands of the government in regards to what is put into the curriculum. This will lead to the garbage taught in the public schools today being taught in the private schools tomorrow.