A traditional stumbling block (though not the only one) is the question of religious schools. Voucher proposals have historically been seen as ways to get the government to fund religion, leading to various backlashes dating back to the 19th century: http://en.wikipedia.org/wiki/Blaine_Amendment
A solution could be to only allow vouchers to be used for secular schools, but that runs into opposition from religious conservatives, and it's also unclear if it'd be permissible to exclude all schools run by religious organizations from an otherwise generally available program. I suspect conservative support for school vouchers that allowed religious schools would take a huge nosedive as soon as Islamic schools started taking advantage of them though, which is currently what's causing the Netherlands to rethink its voucher program.