My ex claimed our son wrongly on his taxes and now mine are being rejected. He says he has to wait until a special amendment period to refile. Is this even a thing?

Someone else claimed my dependent and my return was rejected

If someone else claimed your dependent already, then your e-filed return will be rejected.  Print, sign and mail your own return which claims your dependent.  Mailed returns are processed differently by the IRS. 

You do not have to wait for your ex to amend his return--it takes 3-4 months for an amended return to be processed.  But your child's SSN is now in the system for this year as having been used.  So you cannot e-file this year.

When you mail a tax return, you need to attach any documents showing tax withheld, such as your W-2’s or any 1099’s.  Use a mailing service that will track it, such as UPS or certified mail so you will know the IRS received the return.