While it's not exactly a vine (it doesn't climb), I'd suggest Hoya carnosa, wax plant. It does produce long slender vine-like stems and is often twined around trellises or obelisks to provide support. It produces clusters of star shaped, fragrantly scented flowers but needs some maturity before this happens. It is a very easy to grow houseplant and non-toxic to pets.

The other possibilty is Jasminum polyanthum or pink jasmine. Reported by the ASPCA as being non-toxic. Not as easy to grow as the hoya but a bit showier when in flower.

Not many indoor vines are inclined to flower much, if at all. And the number of indoor plants toxic to pets (cats especially) is much larger than those that are not. There's not a very long list of plants that will work for you.