Wilson Bros Gardens LLC

While there are many plants that can be mildly toxic to dogs and other pets, the seeds and pods of Wisteria can be extremely toxic to dogs when consumed. Not all dogs will eat the pods but I wouldn't suggest taking any chances. It would be better to plant a different type of vine where dogs will roam. 

A better choice of a non-toxic flowering vine for use on a fence is Bignonia capreolata, commonly known as crossvine. I have several different varieties growing and covering a chainlink fence in my backyard where both my chihuahuas spend a lot of time. Another good thing about Bignonia is that in mild climates (USDA zones 8a-9b) the foliage is evergreen. North of zone 8a and the vines might lose their leaves during harsh winters but will flush new leaves in early spring. 

Other non-toxic flowering vines you might consider are:

Confederate Jasmine Vines (Trachelospermum jasminoides) - These are evergreen and produce loads of fragrant flowers in spring and/or summer.

Honeysuckle Vines (Lonicera sempervirens) - These produce loads of fragrant flowers in spring and throughout the season. 'Major Wheeler" is one that blooms nearly year round in my Georgia USDA Zone 8a landscape.