If it is a sole proprietorship, use of an SSN would be acceptable. However, the states I am familiar with would not allow DBA registration in the name of a "Company" because it makes it sound like there is an entity.
If they have no evidence of a DBA registration there is no need to go any further. Even if they did, an existing business generally does not open a new account with checks from its customers, they open an account with a check drawn on another bank.
If its bona fide, they need to prove it to you. Try a Google search on the name and address of "the company..." If it's an entity, call and ask them if they've lost some checks...
_________________________
In this world you must be oh so smart or oh so pleasant. Well, for years I was smart. I recommend pleasant.