Barry, I don't believe that school vouchers run afoul of the First Amendment, and neither does the Supreme Court. In Zelman v. Simmons-Harris, a school voucher case, the Supreme Court explained that its Establishment Clause jurisprudence makes it clear that:
where a government aid program is neutral with respect to religion, and provides assistance directly to a broad class of citizens who, in turn, direct government aid to religious schools wholly as a result of their own genuine and independent private choice, the program is not readily subject to challenge under the Establishment Clause.
Even in the majority opinion in Locke v. Davey, which was joined by Justices Ginsburg, Stevens, Breyer, and Souter, the Court recognized that "[u]nder our Establishment Clause precedent, the link between government funds and religious training is broken by the independent and private choice of recipients."