Almost. They don't have filing status (married/single/etc.) and number of dependents. Unfortunately, those affect your tax rate, so the tax return can't be calculated without it.
In fact you should tune the process - if last year you had a large refund, then this year add more exemptions till your refund is as small as you can make it. (Technically you can go over if you want, and be required to pay, most most people don't like that.)
The easiest way to calculate it is do last years return with varying numbers of exemptions and compare the tax liability number (not the final number which includes payments). Use the real return as a baseline, then keep adding exemptions till the increase in the liability matches your refund, then use that number on the W-9. Perhaps that number minus one.