Here are the file types recognized by Vistaprint:

VistaPrint does not recognize multiple Artboards in Illustrator documents, you can either save your Illustrator to separate files and upload them individually or you can save your file as a PDF and it will recognized multiple pages (PDF files work although they are not listed on the supported file formats list). Here is a screen shot of a PDF test file I created from Illustrator.

If you choose the multiple Artboard option choose the "Save each artboard to a separate file" in the Illustrator Options dialog box (the box that pops up after you click Save).
