Vertigo was boring, confusing, and mostly dumb. The first half was okay, but the rest is pointless and wanders without a definite plot. It had little objectionable material besides being a bit disturbing or intense, but kids and many adults will fall asleep quickly. Probably Alfred Hitchcock's worst movie.
Brilliant, although not for kids - the actual content is all PG-rated, but the film has an intensely dark atmosphere and the themes of obsession would probably go over their heads. OK for mature, smart older kids.
This is a great movie! It is engaging, thoughtful, and features great acting, a great score, and brilliant direction on the part of Alfred Hitchcock. There are some mature thematic elements (deceit, murder, obsession, etc.) and some intense, disturbing images (a woman trying to commit suicide, a woman falling off a belltower, etc).
I don't think the matter at hand with Vertigo is appropriateness, but rather how much your kids will be able to follow... the plot is a bit convoluted. A mature 13-year-old will be able to grasp it, albeit not the incredible, Hitchcockian Freudian undertones and subtleties, but the gist of the film itself.