Should I submit an application to obtain subsidies for high medical expenses?

No application is necessary. The MSD Health Insurance Society checks high medical expenses based on statements of medical service fees which are sent from medical institutions. For those who become eligible for subsidies for high medical expenses, the subsidy amounts will be automatically paid together with their salary into their payroll accounts. 

If a person expects to incur high medical care costs, it can be more convenient to obtain “a Certificate of Application of Maximum Copayment Amount” in advance. By showing this “Certificate of Application of Maximum Copayment Amount” to the medical care institution together with your health insurance card, you can ensure that the amount of medical care costs for which the hospital bills you (per month) will not exceed the Cost-Sharing Maximum Amount and reduce the amount of medical care costs you must pay at the hospital on any single visit. An application is required for the Certificate to be issued.

 
*Insured persons aged over 70 till under 75 can be treated in the same way by submitting the Claimant Certificate for Elderly Healthcare. 　　　　　　　　　　　　But if it dose not incur high medical costs, you will not necessary to apply for the Certifecate.  

But if it dose not incur high medical costs, you will not necessary to apply for the Certifecate.   

See "When you want to reduse the amount of medical care costs you pay at the medical care institution"