Depending on your county in NY there is no state licensing requirement to apply plain fertilizer. If that fertilizer contains any kind of herbicide, insecticide, or fungicide then a 3a applicators license is required. You will have to take the 30 hour course and pass the core and category exams. You will become a pesticide technician which allows you to apply general use pesticides. You then must take DEC 12 credits and get 1 year of experience to move up to a commercial applicator. This allows you to apply restricted use pesticides and apply at schools etc. In addition to an tech/ applicator license, you will also need to obtain a business registration for your license.
Suffolk county requires any business who applies fertilizer to take a 4 hour turf management class. This course is meant to educate business on how to determine the right type and amount of fertilizer to apply in order to reduce nitrogen pollution. Your county may have a similar requirement.
I would start by contacting you local Cornel Cooperative Extension. They will have information on fertilizer laws and licensing. A list of 30 hour courses can be found on the DEC website. I took mine through a local landscape association.
And FYI you can still get in trouble if you spread a fertilizer containing a pesticide when the homeowner supplies the product.