Okay, here's something people say all the time. It's not exactly true. Being a vigilante, itself, is not illegal. However, many of the things that vigilantes tend to do are illegal. As in, The Punisher is an illegal vigilante due to his use of excesive and lethal force. Batman is an illegal vigilante because he engages in breaking and entering when gathering evidence, and has been known to torture, abuse, and otherwise harras suspects when looking for information. But, it is possible to be a legal vigilante. It just wouldn't be especially effective.
Actually, I've figured out how that could work. First of all, Batman is not unnarmed. He just doesn't have any guns. It could work if he did it like this:
First, he wears body armor. The suit should pverall be very protective, bulletproof in vital areas, and have head, knee, elbow, and crotch protection. He uses a large, metal staff, and has a taser straped to his right hand (or whatever hand he tends tp punch with). He drops some flashbangs, and then atacks the thugs while they are disoriented. WIth the taser, he can deliver one punch knock outs, or at least make it so they're so stunned they really can't atack. And being hit by a big metal stick really hurts.