I spoke with Greyhound Customer service on June 21, 2019.
A person without an official ID has one option, which is to pre-pay a ticket online. It has an $18.00 fee on top of the ticket price and is only available in the United States, but it may be your only option.
She told me employees are not allowed to sell a ticket without an official ID, and the debit card won't be enough. The official identification card needs a picture, a first and last name, and an expiration date. It sounds like you may not have that, but, as someone suggested in a comment, a State ID card is a perfect alternative. I'm handicapped and can't have a driver's license, and I use my Massachusetts State ID card for everything!
To order the ticket online, choose the Book a Trip option at Greyhound's website. When you get to the point of purchase and enter the user information, you'll be asked to choose a password. The ticket will be created with a confirmation number. When you use the ticket you need both the password and the confirmation number. (This is designed as a way to buy a ticket for someone else, but it works fine even if it's for yourself.)
From Pre-paid ticket orders:
To pick up their tickets at the station, the person traveling will need to provide:
Photo ID or password chosen by the person purchasing the ticket
Confirmation Number
There are some discounts available to help offset the ticket price. This page has a list.
