There is no public database or other source of employers' EINs.
The employer's EIN is on the W-2 that you received from the employer. If you don't have that, the simplest way to get it is to call the employer's payroll office and ask them for the EIN, or ask them for a duplicate copy of your W-2.
Some employers provide online access to W-2s. If your employer does, you should be able to get a complete copy of the W-2 there.
You can get a copy of your W-2 from the Social Security Administration. There is a fee of $126. See the following link.
How can I get a copy of my wage and tax statements (Form W-2)?
I assume that what you got from the IRS web site is a Wage and Income Transcript. If so, it does not include the state information on the W-2. If you live or work in a state that has state income tax, you need the state information.
In the future, save the W-2 that you get from your employer.
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