As a U.S. citizen, you can file Form I-130, Petition for Alien Relative, to sponsor your spouse for a green card. However, your spouse would need to depart the U.S. and apply for an immigrant visa at a U.S. embassy or consulate abroad 

If your spouse is eligible for a Form I-601A waiver, they may be able to apply for a waiver of the unlawful presence bar without leaving the U.S.

Yes, you can sponsor your undocumented spouse for a green card through a family-based petition, but the process may involve additional steps if they entered the U.S. illegally.