What evidence is required for an insurance contract to be legal?

I don't think that the term "evidence" is the best one to use.

Like any other contract, in order for an insurance contract to be binding and enforceable, 3 elements need to exist: an offer, an acceptance and consideration.

As far as insurance contracts are concerned, the offer consists of the application that the proposed insured submits to the insurer when coverage is sought. The application may be submitted directly to the insurer if it is a "direct writer", or more commonly, through an insurance agent. The acceptance is when the insurer accepts the risk as evidenced by the issuance of the policy. The consideration consists of the insured's promise, and fulfillment of that promise, to pay premiums.