Yes, your father will be able to claim your children as dependents if he files by mail. You will need to mail in an amended return (1040X) to remove your children as dependents from your tax return. This will take some time for the returns to be received and processed by the ITS.

Here is information from TurboTax on this situation from your father's viewpoint:

Claim Dependent

the question is who is entitled to claim them.

if he can claim you as a dependent then it's proper for him to claim the children. he will have to file by mail

so if you are under 19 or a full-time student under 24; lived with him for more than 1/2 the year; and did not provide over 1/2 your support, he can claim you and the kids. 

Will I need to amend mine first before he can send his in? 

No.  He doesn't have to wait to mail in his return.  He should mail it as soon as possible.  Your amended return doesn't have to get there first.  

I am 24 years old, I am a full time student. We live in his home and he supports all of us while I am pursuing my degree. The only reason I had filed a tax return was because I had a 1098-T.

If you had less than $4,300 total income then he can claim you too.