If the adopter does not have a smartphone, it's still possible to do mobile checkout.
In the checkout flow, after checking that the adopter will pay with credit or debit card, you'll click Customer has no phone beneath the field with the phone number.
From there, you have a few options:
1. Show the screen to the adopter to let them complete the checkout process on your staff's device.
2. If your staff isn't using a screen that you can easily share with the adopter (eg. it's desktop computer behind a desk), you can send the link to a tablet, iPad, or nearby computer. Copy URL of the new tab that pops up and paste it into an email. Send that email to an account that you can open on one of your staff's tablets or iPads to hand to the adopter to complete the checkout process.
3. Copy and paste the URL of the new tab that pops up into an email, which you'll send to the adopter to open and complete. The adopter does not have to be with you in person for this option.
Here is a video screencast showing you how to email the link to an adopter:
You can read more about processing transactions here: How do I process adoptions (and other transactions)?