I’m working as an Au Pair on a J visa. My host family would like to sponsor me as a permanent resident. Can they help me stay in the US permanently?
Probably not. An employer can sponsor a foreign national for permanent residence in only very specific cases. The employee needs to meet specific requirements (example: exceptional ability, outstanding professor, or employment that requires a bachelor degree and there are no qualified US workers available for the position. See more here.
Also, as a J1 visa holder, you may be subject to the two-year Home-Country Physical Presence Requirement based on section 212(e) of the Immigration and Nationality Act.
You may have other options for remaining in the US. You should consult with an immigration attorney who could review your options and history and goals in detail. Good luck!