My employer asked me to fill out a W9 because they reimbursed me for an amount greater than 600.00. Is this normal?

If you are an employee, it is not normal.  A W9 is asking for your Identification number which, for an employee, is your social security number.  Your employer should already have it.

It sounds like your employer is going to treat you as self-employed for reporting some expense reimbursements, which usually is not correct.

Best course of action:  ask your employer why they are asking for one and what they intend to do with it.

I asked why and their answer was any time they reimburse for more than 600.00 the IRS requires they have this form on file.

any time they reimburse for more than 600.00 the IRS requires they have this form on file.

Nonsense.  If

• they reimburse you under an accountable plan (they reimburse you directly for actual expenses or at the Government per diem rate) then they don't report it to the IRS.  If you are an employee and reimbursed under an accountable plan, the amounts should not be included on your Form W-2. An accountable plan is described in IRS Publication 463 here. 
• If they provide you with a flat allowance for expenses (a fixed expense allowance) regardless of what is incurred then it is wages that go on your Form W-2 and are subject to Social Security and medicare.

In neither event is a Form 1099 (and related Form W9) necessary, because you aren't self employed.

On the other hand, arguing with your employer is like beating your head on the wall - it generally is futile.  Just be prepared for the complication of receiving a Form 1099-MISC at year end for expenses that don't belong there and having to report them on your tax return.