I was many years at an OEM, in tech support engineering, and I'm not aware of any requirement to notify the OEM when someone applies for an STC. You have to remember that many STCs are for airplanes who's manufacturers disappeared long ago, so you really can't make that a basic requirement. It's possible however, that the FAA may request OEM input like an NTO for a proposed change on a case by case basis, depending on the scope of the mod, and if OEM cooperation is not forthcoming, it could stymie the STC project.
Certainly there is no requirement for the OEM to support an STC for continuing airworthiness, except perhaps where an STC causes an incident or incidents that generate activity toward an Airworthiness Directive and the OEM's Airworthiness Engineering Department has to get involved. Operators use to come to us for help with issues on a subsystem installed under an STC, and we'd tell them sorry, we have nothing to do with it.
An OEM that is still in business may, or may not, participate in development of an STC. One of the difficulties for the STC developer is obtaining engineering Source Control Drawings, which the FAA may require as part of the STC data package. However, a lot of the time the OEM will want nothing to do with the STC, because it implies (to the OEM's customers) that the OEM will be obligated to help support the mod.
The linked Advisory Circular covers the STC application process and there is no mention of No Technical Objection letters from OEMs or requirements for OEMs to support STCs. OEMs will sometimes discover STC mods done on their fleet that they had no idea existed until they start to get visibility in service.
STC Application Advisory Circular