Yes and no. Yes, if you do the research for exactly what kind of soil it needs, you can grow grape vines in pots. BUT this is like asking, "Can I move small trees around with the fences?"
Grape vines are thick-trunked plants that really do require something to lean on. They don't merely lean, they grow around. You need something sturdy to grow them on and once they're grown, they aren't moving, unless you're planning on moving the fence with them.
There are a number of places they could grow? Then find the absolute best place grape vines will thrive and plant them there. (There's a difference in micro-climates in yards. In the northern hemisphere, the south side of the house gets the best sunlight. The west is next, north is worse, etc.) Also, grapes are a fruit, so it might need acidic soil or alkaline. The reasons I don't grow tree size fruit in my container garden is lack of space and too much effort to make the soil right for them.