Depending on the age of the grapevine, and it sounds like it could still be young since you say it’s “small,” it is indeed possible to transplant grapevines. It takes a lot of care and often takes a lot of time and elbow grease, however.
Grapevines quickly establish very deep, wide-reaching root systems and the older they are the bigger those root systems become. When you do dig out the vines, be sure to initially dig at least three feet away from the stump in order to find the rootball, dig around and under it, and move as much as you can of it. Be sure that the hole into which you move the transplant is similarly large — you may be digging a lot longer, deeper and wider than you had planned on. The roots develop many little “micro hairs” and complex and very thin root pieces that are critical for the plant’s health and are often disrupted during the move. When you do get the rootball back underground, be sure to water much more than you normally would for at least three weeks. The vine will be stressed and will be in need of extra care.
I definitely recommend waiting until late winter or early spring to attempt a transplant. At that time of the year (depending on where you live, probably around January or February), the vine will be in its most hibernation-like state. It will have had a full drink of winter rains, a replenished nutrient supply and will essentially be taking a long winter’s nap in preparation for the upcoming growing season. This is indeed why we prune grapevines in the late winter, when we effectively remove anywhere from 40–70% of the vine’s material. July and August are prime growing times for the plant, and if you knock down the vigorous green growth of the vine this time of year, the vine might not spring back. Summertime heat and dry conditions could also further stress a struggling vine, adding insult to transplant injury.
Luckily grapevines are relatively inexpensive to purchase in the case of a less than successful transplant attempt. I do fully understand the emotional attachment we can sometimes have to our grapevines, especially plants that we’ve made great wine from or have moved with us in containers from house to house.