What is the process of immigrating a sibling who is living in the U.S. on H4?

If you file an I 130 for her now, it will be years before a priority date is current. You can review the Department of State Visa Bulletin. If her spouse's employer petitions for an employment based immigrant visa for him, that may be another, better, option.

The Immigration Law provides a visa category for Brothers and Sisters of Adult U.S. Citizens. Unfortunately, there is a limitation on the amount of visas issued in this category that is shared with most of the countries in the world (excluding China, Mexico, India and the Philippines which have their own visa limitations). You would need to file a family petition for your sister as soon as possible in order to give her a "number" in the visa waiting line. The wait time is determined by the Department of State and mainly depends on the amount of visa numbers requested in that category. You are able to see the current visa wait times in the DOS website under the "visa bulletin" tab. The visa waiting line could be up to 11 years depending on her country of nationality and she may not wait for this visa number in the U.S. if she is "out of status." I recommend that you visit a qualified immigration attorney in order to discuss different avenues that your sister could take in order to remain in lawful status in the US or to obtain a green card through another type of sponsorship. Our office limits its practice to Immigration Law and we would love to assist you and your sister through this journey.

No she cannot just stay in the us just because the I 130 may be approved. She must maintain her non immigrant status. It takes about 10-12 years for a visa priority date to become current for a sibling of a us citizen and she is not able to submit the adjustment of status application until a visa number is available. She should probably have her husband explore employment based options for permanent residence while there is still time. If a PERM application is filed by his employer before the end of the 5th year in h1b status, he will be eligible to extend the h1b status beyond the 6th year maximum for as long as necessary to complete the permanent residence process.

It will take more than 10 years before your sister can get her green card through you. She cannot stay in the US because you have filed a I-130 for her (and family).