The 24-year-old former Sheffield Wednesday and Chesterfield shotstopper spent a spell last season on loan at Stockport and is now looking to establish himself as the Saddlers' first-choice.
"Richard is a lad who has been with us for the last few weeks," manager Dean Smith told the club's official website. "He has impressed during that time and displayed an excellent attitude.
"I am pleased to have him on board; this is the chance for him to find his way back into regular first-team football.
"He knows that I am also looking to bring in another keeper to provide competition for him and Liam Roberts, but he is relishing the challenge."
O'Donnell added: "It's been a long summer. I've never had to go anywhere on trial before, but the lads have been different class and really helped me settle in here."