I do not believe such information is publicly accessible.
States have business registries. Business registries have a registry number, but won't/don't display EIN or tax identification numbers to the general public. Registry numbers are different from EIN or tax identification numbers. It's unlikely the registry number would help you in any way. There's no "national business registry" as far as I know - at least not from the US Government. There are some privately owned web sites that seem to merely spider state registries. I have no idea if such private sites are worthwhile. But the actual government-run state registries don't display any EINs or tax identification information.
Much like an individual Social Security Number in the US, there are reasons making business EINs public knowledge may be a bad idea. Anyone could fraudulently use the EIN. The first thing that comes to mind is money laundering -- anyone could look up an EIN and fraudulently claim they were paid X amount by that company. Then the company may face some undue hardship for failing to report that payment, even though it was not legitimate.
To the best of my knowledge, an EIN can only be obtained from the company itself - often on a 1099 or W9 form. --- Of course, I could always be incorrect.
-- anecdote ---
The US Internal Revenue Service (IRS) won't even give me my own EIN if I call and ask for it.
When I got my EIN it was faxed to me and was written, by hand, on the form. I had considerable trouble simply reading the number - was that a 6 or an 8, was that a 7 or a 9.... I called the IRS and asked, they would not provide the number to me, but if I guessed it they would confirm it was correct or not. So I spent 5 minutes guessing what I thought the fax read... eventually guessing correct.