As per my understanding you want to update column "DateTravelPlanSubmitted" of the table "Schools" whenever data of "SignOff" table is updated.
If this is correct then you can create a trigger on update
CREATE TRIGGER Trigger1
Update Schools SET DateTravelPlanSubmitted = [Your Value]
WHERE Schools.PremaryKey = inserted.Primarykey
Please let me know of you have different requirement then I understood or you want more information.