My son is 4 years old and receives disability payments. Do I need to file taxes for him? Do I report it on my own taxes somehow? Should I have received some kind of form?

Do not enter the SSA1099 for your child on your own tax return.   Someday if the child has other income  (after school job, etc.) and is filing a tax return, for that tax year he will need to include the SSA1099 on his own return.  Until then, it does not get entered anywhere.

No. You do not have to file a return for your son or include his disability payments on your income tax return.

Supplemental security income (SSI) payments are not taxable.

See Regular & Disability Benefits.