"The employees mentioned below have been promoted to new designations . Hence, we suggest that health cards would not be required for them. Could you please look into this and do the necessary / as suggested "

"Position" instead of "designation". "Thus, we suggest that the health card not be required for them." (I assume "this staffs" refers to the employees from the first sentence...?) "Can you please look into this and do what's needed?" or, less formal, "...do what you need to do?"