I believe you're required to provide a tax ID number, not a social security number. For US Citizens opening personal accounts, these are the same. Foreign citizens can apply for a tax ID number from the social security administration (there are forms online somewhere - or the bank can provide one). We have used foreign Au Pair's for child care, they have never had issues opening a US bank account using a Tax ID number.

Also note, the banks will usually open up an account with an "applied for" notation. I believe they give you about 4 weeks to supply the actual number.