On signing the deal, Pickering told the official FGR site: "I'm excited to be here – there's good coaches and good players so I'm looking forward to getting started.
"I know I've got a lot to learn this year and I've got all the tools here to do that - working with the staff, Brad and Sam should help to make me a better player by the end of the season."
Goalkeeping coach Steve Hales added: "We had a few lads on trial here recently but Harry stood out as he trains hard, listens well and has a lot of potential to develop.
"He'll share the workload with Sam and Brad and I'm really looking forwards to working with him."
Pickering, who failed to make a senior appearance for Vale, spent time on loan with Kidsgrove Athletic in England's eighth tier last season, and was among 13 senior players to leave Vale Park in May.