Could you give me a medical certificate to submit to my insurance company?
Literal
Insurance-company [to-へ] submit [for the purpose of-のための] medical-certificate [obj-を] please.
~のための is the attributive form of ~のため ('for the purpose of X') modifying a following noun. Literally 'the medical certificate which is for the purpose of submitting to the insurance company.' 診断書 is a specific medical form — a doctor's written diagnosis, used for insurance claims, legal purposes, and workplace absence.