PC Gamer reports that Fallout 76 will have private servers but it looks like they will be online only.
More information.Will Fallout 76 get private servers?
Yes! At QuakeCon 2018 on August 11, game director Todd Howard confirmed again that private servers are part of the plan. "That is definitely something that we are doing. We are committed to it. It's not just having a private server, it's being able to mod it," he said in a Q&A panel. "It's something that, given the online nature of it, is going to be very, very complicated, but we're committed to it and we've been starting to design what that system is going to look like. It's a complicated problem but one that we're one-hundred-percent committed to solving."