I have not done anything like this, but the USPS has " USPS Operation Santa"[0] so kids send letters using the actual postal service and people can adopt the letters.(I have no idea if it works outside the US, I think it maybe?)
Maybe something like this if you want to keep it local.
For example. If you have some type of networking or regular meeting for the neighborhood already, ask for volunteers to reply to letters dropped in a community "mailbox". Make it a letter reply "holiday event" for the community, where people gather at a pot-luck/food and a lot of pens and Santa themed stationary. As long as you trust those in the neighborhood to not troll some kids...
[to add] I would guess parents that take the kids to the "mailbox" to drop off the letter would be willing to take the time to pick up the letter at a cental location, or allow a delivery address to be included that you or someone could drop off at front doors, maybe on Christmas eve? But I personally hesitate on that part because I dont know people in my community, hardly at all, and I am paranoid like that and not that kind of trusting
[0] https://www.uspsoperationsanta.com/how
