Re NY P-EBT
For children not on SNAP Seems as your schools need to try matching you up on an online portal with a Medicaid case if they find one.
If they don't have on they mark it as other. Those will need to wait for a card in the mail.
If you don't have SNAP but have Medicaid you can try calling the school to help them match you up so you will get the money on to your Medicaid card