It is my interpretation of their wording that an actual physical paper check will be sent. I do not know if it will be sent to the consumer's mailing address or in the box with the replacement crank to the dealer.
Turn around time is also not known to me at this time.
Make certain that all of your contact information including your mailing address is up to date with your LBS, just in case Shimano intends to mail a check directly to you.
Cheers!