I do not believe so as even permanent residents have hard time adopting if they are not citizens. But I think it's best if your consult with local adoption attorney. The issue will be that any non citizen status implies you may leave the country, so, what will happen with the baby?
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There is not specific requirement that a person have any particular immigration status to adopt a child. I have represented undocumented people in Santa Clara County in adoptions proceedings who were actually being deported, and had no trouble.
There is a law that says that immigration status cannot be considered in determining whether to place a child in the foster care of a relative.
I am not 100% familiar with the rule for foster parents who are non-relatives. However, I do have one client who is in removal proceedings who became a licensed foster parent and then adopted the child.
Foster parents are not paid an income. They receive money that is the property of the child for the child's maintenance. So, it would not implicate any restrictions on an H1B visa holder working for a particular employer.
I think that the temporary nature of your immigration status would be one factor that would be considered, but probably on balance would not be enough for the to deny you participation as a foster parent or an adoption.