Answer: One consideration might be that if the arborvitaes never grew well due to lack of light (this is possible, they do prefer full sun), adding the fence will further reduce the amount of light they receive.
Here are some vines that will grow well in shade. English ivy is an evergreen vine that will climb any type of fence. Virginia creeper is a deciduous vine that will also climb any type of fence. Both of these use holdfasts and will mar the surface of the fence if you ever decide to pull the vine down. They may also need regular trimming to keep them under control.
Fall blooming sweet autumn clematis will grow and bloom in quite a bit of shade. This vine is deciduous and can be cut back as needed in early spring to keep its size under control and to promote blooming. It will twine through the fence initially and then will scramble along the top of it. To get it to the top of a solid fence you can support it with a small section of trellis. Once at the top it will manage for itself.
Climbing hydrangea (H. anomala petiolaris) is a slow growing but elegant vine you might consider. It uses holdfasts to climb so does not require a trellis.
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