Two different questions get asked with almost the same words. One is going from a business name to its number. The other is going from a number you already hold back to the business it belongs to. They have different answers, and the second one is the harder of the two, so it is worth separating them before starting.
From a business name to its number
This is the ordinary direction. It works when the business has disclosed the number in a public filing: securities filings carry it on the cover page, exempt organisations have it published with their annual return, and federal award records usually carry an identifier. If the business is a private company with none of those, the number is not public and the search ends with a direct request to the company.
From a number back to a business
Going in reverse is harder because it needs an index keyed on the number, and no public index of that kind exists for US companies generally. Where the number appears in a published dataset, such as the exempt-organisation lists, a reverse match within that dataset is possible. Outside those datasets there is nothing to match against, so a reverse search on an arbitrary nine-digit number will not resolve.
What the state register settles either way
Whichever direction you are going, the state business register is what confirms the entity is real, active and named the way the paperwork says. It returns the legal name, the file number, the entity type, the status, the formation date and the registered agent. For a counterparty check that is usually the substance of the question, and the federal number was only ever a proxy for it.
Questions people ask about how to find a business ein number
Can I find a business by its EIN?
Only within a dataset that publishes the number, such as the exempt-organisation lists. There is no general public index keyed on the number.
Is a business EIN the same as its state file number?
No. The EIN is federal and issued by the IRS. The state file or charter number is issued by the state register and is a different identifier.
Which is easier to find, the name or the number?
The name. Every state publishes its business register by name. Federal numbers are published only where a filing required it.