Can you adopt an immigrant without a green card?

Yes. According to alllaw.com, it is possible, but difficult, and can only happen under very specific circumstances:

  • The person would have to be a child.
  • The child would have to live with you, in your legal custody, for two years before you file the visa petition.
  • You would most likely have to be a U.S. citizen.

The process is complicated and will probably require an immigration attorney.

(see alllaw.com/articles/nolo/us-immigration/adopt-undocumented-illegal-immigrant-get-green-card.html)