No Person except a natural born Citizen, or a Citizen of the United States, at the time of the Adoption of this Constitution, shall be eligible to the Office of President
You can argue that it doesn't say anything about vice president, but then he can't be part of the line of succession so what is the point. Though I wonder if maybe saying he can't be vice president on those grounds means speaker of the house also must be natural born... The constitution is short and so you can come up with weird situations like this all over.