If the third party is an official dealer, they should be authorised to do warranty repairs.
Yes, they should be authorized by the manufacturer. Though only Eyetech knows who that is. Eyetech should be providing the authorizer retailers/resellers with a back channel to the manufacturer. They should be trying less to be the front man of a manufaturer they got to produce the boards and more of a distributor.
It might be a better idea for them to separate the two, as they seem to want to be a manufaturer but operate as a distributor at the same time. They may need to separate the A1 production bussiness from their retail/distrobution business. Have a separate business or unit that licences and outsources the production of motherboards and another unit with is just product distrobution/retails sales. They could have Eyetech Industries Ltd. and Eyetech Distrubution Ltd both under Eyetech Group Ltd. I think it would bring clarity of purpose that would help make things work more smoothly.