Materials suppliers are not normally considered subcontractors, and are not required to undergo subcontractor approval or submit payroll information to ADOT. However, you must be careful not to perform any action that is a part of the construction project itself or you will be considered a subcontractor. For instance, an asphalt supplier may haul asphalt to a construction site, and even load the laydown machine, while remaining a supplier. Were the asphalt supplier to then operate the laydown machine, however, it would be considered a subcontractor operating on site without approval.
Provided you act only as a supplier, obtaining jobs with contractors working for ADOT is very similar to obtaining jobs with contractors working on private sector projects.