Thank you for your question about your vine's identity.   I believe it is a Passiflora suberosa, or corky stemmed passionflower.  I've attached an image of a positively identified one, with leaves and fruit.  You should have seen rather dramatic flowers, which bloom all year.  Here is a link to an article about this plant, which is native to the Southeastern U.S.  As to whether the fruits are edible, I have found conflicting views on the subject, none of which do I deem to be reputable enough to advise one way or the other.  (Most seem to agree that they contain small amounts of cyanide.)  Before you consume them (or allow your pets to), I suggest you either contact your local Extension office, which may have better information than I have been able to access, or your local health department.

Hope this is helpful.  Good luck!