If an employer is required to have certified emergency responders on site per an applicable standard (e.g., permit-required confined spaces), then training of other emergency responders must be done by an organization that can issue such certifications (e.g., the American Red Cross).
If an employer is not required to comply with one of these standards, then OSHA does not specify that the trainer be certified.
As such, emergency responders can provide first aid training for all employees. Employers may wish to consult a legal professional, however, regarding the potential liabilities created by not having designated emergency responders certified by a recognized organization.