I have the iPhone 5 and I wish to purchase a new iPhone. Is there another avenue to obtain the Apple Card?

Currently the only way to apply for an Apple Card is using the Wallet app on an iPhone. See—>Steps to take for applying to Apple Card - Apple Community.

Correct, Apple Pay requires 12.4 which is not available for the iPhone 5. But if you want to buy a phone using the Apple Card 0% financing an Apple store may be able to help you. Stop in and ask, or call Apple. The number will be on the website for your area.