You cannot sponsor Green Cards for your in-laws however your spouse can sponsor them if he/she is a U.S. citizen and 21 years of age or older. If you are a U.S. citizen, you can sponsor Green Cards only for your spouse, your parents, your children, and your siblings.
If your spouse is a Green Card holder he/she cannot sponsor their parents as it is applicable only to U.S. citizens. So if you are looking to get your in-laws to get a Green Card then you should help your spouse become a U.S. citizen. It could take less than 2 years to become a U.S. citizen after applying for it. Even if you are a Green Card holder and your spouse is eligible for citizenship, they should apply for citizenship so that they can sponsor Green Card for their parents aka your in-laws.
The availability of immigrant visas is unlimited for parents of U.S. citizens as they fall under the immediate relative category. So there are no long wait times involved which is a plus considering a lot of people cannot get Green Cards as quickly as possible since they fall under other categories.
General process to sponsor Green Card for inlaws
As established before you cannot sponsor in-laws even if you are a U.S. citizen. So only your spouse can sponsor his/her parents to get a Green Card. If your spouse is already a U.S. citizen this is the following process through which your inlaws can get a Green Card.
Your spouse should file Form I-130 to sponsor his/her parents. Separate Form I-130 should be filed for the father and mother. You cannot file just one form you should file separate Form I-130 for each parent.
After filing Form I-130, if the petition is approved then the next step will depend upon where the parent lives. If they are within the U.S. legally then they may be able to apply for Form I-485 to adjust their status to Permanent Resident.
If they are outside the U.S. then they need to follow the consular processing process. Their application will be sent to the National Visa Center and once Form I-130 is approved they have to fill out Form DS-260 and DS-261. Once approved they will be called for an interview at the nearest embassy and after the interview, if a visa is granted they can travel to the U.S.
After landing in the U.S. Customs and border protection officer will inspect the person at the port of entry. After approval, they can enter the U.S. and officially become Permanent Residents. They will most likely get an I-551 stamp which will act as proof of permanent residency until the actual Green Card arrives. USCIS will mail the Green Card to the mailing address given and they can enjoy their life in the U.S. with you and your family.
If you would like to see other options connect with an immigration lawyer to discuss them.