How can I get an SSA 1099 for my minor dependent children so I can get money I had withheld refunded to them?

Please stop the withholding going forward.

Did the child get their SSA-1099?  Or do you need to get one from Social Security?  How to get another copy,
https://faq.ssa.gov/link/portal/34011/34019/Article/3706/How-can-I-get-a-form-SSA-1099-1042S-Social-Security-Benefit-Statement

Then the child will have to mail in their own return to get the withholding refunded.  If they only got SS or SSDI it is not taxable to them and you can't efile a return with no income listed.  So they will have to mail it.  

The Online version is only good for 1 return. But you can use the same email address for 5 accounts.  To do another return you have to set up a new account and new user name and pay any fees again.

How to start another return in the Online version
https://ttlc.intuit.com/replies/3288574