Unfortunately, he doesn't seem likely to risk any of his political capital on that. A search of his official web site on the word "gun" doesn't return a single hit that even discusses gun control, or the proliferation of guns in Virginia. The strongest statements by him that I can find in newspapers imply that he might be willing to support laws making it more difficult for those officially ruled mentally ill to purchase guns (see e.g. this article in the Washington Post.) According to Media Matters when Kaine was asked if he was "concerned" about the fact that the Virginia Tech gunman had access to a gun that at one time would have been outlawed by the federal assault weapons ban, he replied "Dealing with families is first. The careful and independent assessment of what occurred is second. Once that is done, there will be ample time to discuss whether there need to be any changes made to policy here or elsewhere."
This doesn't sound like a man anxious to take on gun control.