The Yorkshire firm replaces LDV Vans, who went bust just before this year's final.
Clubs in League One and Two challenge for the trophy, along with top-finishing Conference sides.
The seven-figure deal is said to be the biggest in the competition's history.
Football League chairman Brian Mawhinney said: "All supporters want to see their club competing for silverware and the Johnstone's Paint Trophy gives clubs from Leagues One and Two a cup competition that realistically they can win."