Hi Rob van Olmen,
When setting up and using Apple Pay on your Mac, make sure you have Touch ID and that it is set up and that the website supports Apple Pay as well. If so, see Use Wallet & Apple Pay on Mac.
"Set up Apple Pay on your Mac
- On your Mac, choose Apple menu > System Settings, then click Wallet & Apple Pay in the sidebar. (You may need to scroll down.)
- Click Add Card.
- If you’re not signed in with your Apple ID, you’re asked to sign in. After you’re signed in, return to Wallet & Apple Pay.
- Enter your payment information.
- Use a card on file: If you have a credit or debit card on file with Apple for making purchases from the iTunes Store or App Store, enter the security code when asked.
- Add a new card: Position your card in front of your computer’s camera so that it appears within the frame shown on the screen, then follow the onscreen instructions. Or click Enter Card Details Manually, then provide the requested information. Repeat for each card you want to use with Apple Pay.
Important: If you sign out of your Apple ID account, you have to set up Apple Pay again—including adding any cards you’ve added in Apple Pay on your Mac."
Additional resources that might help:
Take care.