The Bantams were hoping to bring in the 32-year-old Charlton Athletic striker only for a problem to arise in his medical.
City, who had agreed a £100,000-plus fee to sign Benson in January after initially taking him on loan, asked to restructure the deal but were turned down flat by the South London club, who instead opted to loan Benson to Swindon Town.
It meant Thursday’s deadline passed without manager Phil Parkinson being able to add to his squad.
Joint chairman Lawn said: “It was a frustrating few days. We tried our best but couldn’t get any further in our attempts to get him. It was disappointing after all the effort we had gone to.
“The money also wasn’t something the club had, it was extra money that myself, Julian (Rhodes, joint-chairman) and his father (Professor Rhodes) were willing to put in to allow the manager to bring in a proven quality striker, which is something we haven’t had since Peter Thorne left.”
Parkinson’s desire to bring in a striker is down to the current crop having failed to fire this season. James Hanson has five goals to his name from 15 games but City’s next highest scorer in the league is midfielder Michael Flynn with four penalties.
Lawn added: “Us not being able to bring a new striker in this week gives a chance to the others. If they can perform in the next few weeks then it means we won’t have to go out and bring someone in when the window re-opens.
“Strikers are judged on goals and we need to get more from ours. It is in the interests of our existing strikers to score because if a new striker does come in then the chances are one of the others will have to go out.”
City are third bottom of the Football League and have lost five of their last six visits to Gillingham, but Lawn insists the team can improve that record today.
He said: “We have shown in the cups what we can do; we need to take that into our league form.”