Would it be possible, if I kept it pruned, to keep a grape plant in a container? I have 23 inch containers that I could use to keep them in, but would that be enough? What would I need to consider if it's possible?
You can grow grapes in containers. But the smaller they are, the more time and energy it will take to get them to flourish.
Smaller containers mean repotting and root pruning more often. In a large container, say, 4'x4', this will not be necessary on a regular basis. Your 23" containers are rather small, and yearly root pruning will be best in that case. Grapes naturally grow very long viney roots, and will circle the pot. In spring, when pruning the vines, it will be highly beneficial for you to remove the plant from the pot, remove the potting mix, and prune back the roots about as hard as you took back the top, taking out all the long roots. You can then add new potting mix.
Don't expect very big vines from potted plants. They shouldn't be allowed to get too large, or the root system will be crowded. Another issue is that the support will need to be able to move with the pot, so it won't work well with all support systems. Also, potted grapes don't like their root-balls frozen solid too often, and it can kill them, especially if their mix is wet. I've seen this happen to a patio gardener who forgot to protect the pots.
I think that grapevines are a small enough investment that it's generally better, if possible, to plant them in the ground, and count on starting anew if you have to move. However, if you foresee yourself moving in the next 5 years or so, you may want to try to keep them in containers for that period of time, and taking them with you.
Yes it is possible. The biggest hurdle would be keeping it trimmed to a manageable size. The link below has some good information.