Can the mother of the child I am sponsoring for the green card, apply for a tourist visa through the child has the green card?

If you are filing a petition for the step child, then arguably you are already married to the step child's mother, and if that is the case you can file a green card for both of them simultaneously. Any attempt by the mother to get a tourist visa while you are applying for a green card for your step child would be looked upon as immigration fraud, because obviously the mother would most likely want to remain with her child once she arrives in the United States. You should discuss all the facts of your case with counsel in a private phone consultation