What can I do if the other parent claimed children when our legal agreement states I am to claim them?

[JulieR]

You may still file your return and claim your children as dependents but, because the other parent has also claimed them, you will not be able to e-file your return. You will need to print out the return and file by mail.  (see How To Print & Mail A Return)

Tax law permits only one taxpayer may claim any one person as a dependent on a tax return (except, of course, in the case of a married couple filing jointly). If you file your tax return and someone else has already claimed your dependent, then the IRS will apply the tiebreaker rules. IRS will send a letter to both parties claiming the same dependents that may request additional information.  

The rules for claiming a dependent are detailed below. https://ttlc.intuit.com/replies/5187060