@ckumark ,
There are mainly 2 basis for applying for Permanent residency in US (i.e. Green card)
1. The employer needs you in US permanently since they cannot find qualified employees. They would file a LCA (Labor Certification Application) that proves they tried searching for employees with certain qualifications but could not find. They comes I-140 application that proves you meet all the requirements and need to stay in US as LPR. When your date becomes current (if you are from high immigration country) you can apply for I-485 (AOS) to change status to LPR.
2. You have family member who is US Citizen or LPR and needs you permanently in US. They file I-130 (Petition for Alien Relative) to prove they can sponsor you and to prove the relationship between you. You can do AOS after I-130 is approved.
In your case if your basis is employment based, someone would have to do the LPR and I-140 process to show the employer tried searching for local talent including putting ads and searching new employees by all channels but couldn't find suitable person and you meet the needs. I have not heard of an individual doing this on there own without company's involvement. But I am not a legal professional. Maybe it can be done.