Yes, you may have a seedling grape vine coming up in your yard, but my seedling grapes more often come from bird poop than squirrels. A neighbor has a Concord grape vine, which is enjoyed by birds as well as rats and squirrels. The chances of a seedling producing delicious grapes are remote, but a possibility. However, if you want a producing grape vine, you should plant a variety you like.
Are you in an area accessible to raptors like hawks and owls? Sounds like you could use some help in critter control. A CooperÂs hawk and a Barn owl have adopted my backyard, as well as all the neighborhood cats- in spite of my three dogs.
I have a grape seedling that has been sharing a 15 inch pot with another plant the last few years. Guess itÂs time it had its own pot, probably a 5 gallon sized one. The nurseries have them growing in pots for years.

We often have native grapes starting in our garden and we treat them like other weeds, pull them as soon as seen. Al