Since you paid child support for the child, the department that administers that program should have the SSN on file. You'd probably have to jump through some hoops to get it, because of confidentiality concerns.
If you had your child's Social Security number on a previous tax return, you can find their SSN there.
You can contact the Social Security Administration using contact information at the Social Security website, but the information wouldn't be available online.
Neither the Instructions for Form IT-209 nor the Department of Taxation and Finance's Noncustodial parent earned income credit page provide advice on what to do when a custodial parent refuses to provide a child's Social Security number.
You can try to contact the Department of Taxation and Finance for direction, but they are probably overloaded in this time of Coronavirus.
Assuming that you're not able to get an answer, you may need to paper-file your return with an explanation of why you don't have your son's SSN. The tax department may or may allow the credit straight off, but will surely have procedures that cover the situation.
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