Defenders Jores Okore and Antonio Luna, midfielders Leandro Bacuna and Aleksandar Tonev, plus Danish striker Nicklas Helenius have already arrived at Villa Park.
Former Norwich boss Lambert was also keen to strengthen his goalkeeping ranks, and has returned to his old club to secure the signing of 20-year-old Steer, who will complete the move to Villa Park once his deal at Carrow Road expires.
Steer, who has featured for the England Under-19s, will provide competition for Brad Guzan and Benji Siegrist with Shay Given expected to leave the club this summer.
Lambert told the club website: "Bringing in Jed will add to the goalkeeping department and helps with everybody pushing one another, learning from one another and making everyone better and wanting to improve.
"You want competition for places throughout the team, which is something we didn't have last season."