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 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](https://www.greyhound.com/en/) option at Greyhound's website. When you 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. 

From [Pre-paid ticket orders](https://www.greyhound.com/en/help-and-info/ticket-info/cardholder-not-traveling):

 > 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