Any device that is compatible with the Apple TV+ streaming service, can be used for this free weekend. That includes third party devices and browser access.
A payment method shouldn’t be needed for this free streaming weekend. Accounts without payment method can be used for this weekend. A payment method linked to your account would be needed for accessing Apple TV+ when used by subscription, including regular free trial, or promotional extended free trial.
You will need an Apple Account with all the other user information, including name, DoB, country (state), and a verified email address (reply to test message). Payment method can be skipped for now.
How to create a new Apple Account - Apple Support
Add a payment method to your Apple Account (later) - Apple Support
Access on the web may ask for payment method details as a way to set parental controls. Not sure if that is what is interfering here. Parental controls for Apple TV+ on the web - Apple Support. The adult account holder with the payment method should be the one that sets any age restrictions (all on the web, any supported browser, any supported web device) if wanted. It is handled differently on devices with the Apple TV app (per device).
Get the Apple TV app or Apple TV+ app on your smart TV or third party streaming device - Apple Support
Watch on these smart TVs and third party streaming devices - Apple Support
All the ways to watch Apple TV+ and more - Apple Support