I'm of 2 minds on this.
Negatives: I don't like one bit that you're being pressured into "all the other kids...poor, poor me" at any age, but especially as adolescents want more and more responsibility (unsupervised parties, automobiles, you name it). Kids naturally choose kids with whom they have something in common. My kids asked about toy guns at a young age, and we had a discussion that I would consider it if/when he could explain to me why shooting another person was entertaining and a worthwhile goal. He never could come up with one sentence. We have a massive gun problem in this country, and normalizing it so that kids don't react to seeing one worries me. I worry if guns are at all realistic-looking, and what that means if someone else with a gun, or a police officer, comes upon them. People have been shot with toy guns and with cell phones, and not when they're in a crowd. I worry about what they will want to "upgrade" to when this is too boring. I am concerned that they promise to be careful and wear eye protection, and then they invite other friends in who aren't trained and have no protection, and someone loses an eye.
Positives: They're playing outside and getting exercise. They have a group and he's not moping alone at home.
So, is gun play the only way to achieve this? Obviously you need to learn more. You can go to a store, yes, but the manager's job is to sell you a gun and convince you that it's a swell idea and harmless. You absolutely have to have serious consequences for even a minor breach of safety - not wear eye protection for even 5 minutes, etc. If you do a serious safety course and he's still interested, and if he's working a job or doing neighborhood chores to pay for this, maybe it's okay. If you've scoped out the woods and there are clear boundaries with neighboring houses and zero chance that younger kids will hear the shouting and come to explore, maybe. Do talk to your local law enforcement (just go down during regular business hours) and ask if there are problems, if there are town regulations, and if they have suggestions. If they hate this, say no - you do not want your son on police department "radar." Kids need to learn that they don't get everything they want. And you have to follow your gut. Next he'll be telling you he wants to stay out until 1 a.m. and he's the only one who isn't allowed to. He'll want to be at parties you wouldn't approve of, but he's "odd man out." Figure out if he's got his act together or if he's drawn to the wrong kind of kids and feels unaccepted elsewhere - depression is on the rise and we don't want our kids thinking guns are the answer to anything.