The 18-year-old had a spell on loan at Northern Premier League Buxton last season, and was on the bench for the FA Cup win at York City, as well as the subsequent second-round defeat to Morecambe.
He also played an important part in Bradford's under-19s winning the Youth Alliance North East title.
“Getting into the habit of success really increases your desire to keep winning, so I think that is something that will really help me moving forward into the professional game," said the academy graduate.
“I cannot wait to keep working hard in training and looking to kick on and hopefully earn something similar again next year."
Having trained with the seniors at the start of pre-season, Richardson has now signed a one-year contract.
Harry Lewis has joined from Southampton to be the first-choice goalkeeper and 37-year-old Colin Doyle has returned to the club as a player-coach. All last season's senior goalkeepers have left the club this summer, with Alex Bass returning to Portsmouth at the end of his loan, Richard O'Donnell released and Sam Hornby making his Colchester United loan permanent on Tuesday.