When does the FAA require OEM involvement for the approval of an STC?
It tends to go on a case by case basis.  OEMs tend to steer clear of STC developers because OEM involvement tends to drag the OEM into an implied commitment to support the STC itself, or support the STC's secondary effects on how the airplane is operated, that the OEM would rather not be stuck with.

What will happen is an STC developer will submit a certification application with documentation, test reports and all that, to the regulator, and most of the time the regulator will approve the STC independently of the OEM.  But there may be cases where the regulator's engineering group will demand OEM engineering input, or at minimum an "NTO Letter" (NTO = No Technicial Objection) before approving the STC.

In such a case the ball is in the OEM's court and if the OEM insists on not getting involved, well the STC developer is stuck and the project might die on the vine unless they find some other work-around.

In other cases however, the OEM may actively support an STC application where they would like to see the mod go into service because there is a benefit to the OEM.  You might see this for a mod that improves fleet reliability, or is just really popular with operators, but where the OEM lacks the internal resources to develop it in-house due to budget limitations.