Apple Pay Cash is designed to be used in conjunction with Messages. It has only a person-to-person capability. There is no physical card associated with the service.
Apple Pay Cash - Official Apple Support states: "Sending and receiving money is as simple as sending a message. And managing Apple Pay Cash is easy in the Wallet app."
https://applepaycash.greendot.com/termsconditions/ also clarifies the arrangement:
"The Services consist of two separate functions that we will provide unless we otherwise designate a third party to perform any part of those functions: (i) the "Apple Pay Cash Card", which is a virtual payment card that is issued by Bank or another card issuer that we designate; and (ii) the "Funds Transfer Service", which is a service that allows you to: (a) send money to other individuals that are eligible to use the Services and have accepted these Terms (each such person, a "User", and each such transfer, a "P2P Transfer"), (b) load funds onto your Apple Pay Cash Card with a Supported Payment Card (as defined below), and (c) transfer money from your Apple Pay Cash Card to your designated bank account at a U.S. financial institution (a "Bank Transfer"). Each aspect of the Services, including the conditions and limitations on their use, is described in greater detail below...."