Aerial lifts (including boom trucks) fall under OSHA’s scaffolding standard at 29 CFR 1926.453. That section doesn’t have any requirements for an “operator’s license.”However, in order for an employee to operate an aerial lift they would need to be properly trained. Who conducts this training, and what the training must consist of is left up to the employer.
For construction, 1926.20(b)(4) indicates that, “The employer shall permit only those employees qualified by training or experience to operate equipment and machinery.” Also, 1926.21(b)(2) requires that, “The employer shall instruct each employee in the recognition and avoidance of unsafe conditions and the regulations applicable to his work environment to control or eliminate any hazards or other exposure to illness or injury.”