U.S. citizens can, under federal immigration law, petition for (sponsor) their foreign-born brothers and sisters to come to the United States. They fit into the fourth preference category of family immigration. But what about when a parent has had more than one marriage or relationship, and the citizen thus has half-sisters, half-brothers, stepsisters, or stepbrothers? The answer to whether they can apply for U.S. lawful permanent residence (a green card) depends on various factors to do with family relationships.
A U.S. citizen who is age 21 or older may, by law, petition to have brothers and sisters (siblings) immigrate to the United States in certain circumstances. The definition of sibling in U.S. immigration law includes people who were the children of at least one common parent; that is, it includes half-blood siblings. (See the definition of "child" in Section 101(b) of the Immigration and Nationality Act or I.N.A.).
If you are "paternal half-siblings" (the common parent being your father) you will need to provide documentation to U.S. Citizenship and Immigration Services (USCIS) showing that your father was once married to your mother, that their marriage has legally ended, and that the father has married your half-siblings' mother (and that if she was ever married, that marriage, too has legally ended).
Again, a U.S. citizen who is over the age of 21 can petition for siblings, and the definition includes not only biological brothers and sisters, but step-brothers and step-sisters under certain circumstances.
The key is that both you and your stepsister or stepbrother, at one time, met the legal definition of a "child" of a common parent. It doesn't matter if you never lived in the same home.
That means that one of you will need to meet the definition of "stepchild" under U.S. immigration law (specifically, Sections 101(b)(1) and (b)(2) of the Immigration and Nationality Act or I.N.A, or 8 U.S.C. § 1101(b)(1) and (b)(2).), which requires that the child have not yet reached the age of 18 when the parents' marriage took place.
You should realize that even after preparing and filing the visa petition that starts off the process (on USCIS Form I-130), it will be an astonishing number of years before your half-siblings or step-siblings can complete the process of obtaining U.S. green cards.
Due to high demand, the average wait in the sibling category ("fourth preference" or F4) is around 16 years for most applicants, 17 years for those from India, 23 years for those from Mexico, and 19 years for those from the Philippines (as reported on the State Department's Visa Bulletin.)
The wait could change by the time your siblings are in the system, depending on how many others apply around the same time they do, and how great the backlog is before them. Still, it is almost guaranteed to be a very long wait.
They will not be allowed to live in the U.S. during this waiting time, unless they can convince a U.S. consulate to issue a temporary visa (for example, as a student or visitor). But consulates are reluctant to do that in cases where the person has a family-based visa petition on file, fearing that the person will misuse the visa in an attempt to stay in the U.S. permanently before they've got a green card.
Another concern is that the U.S. Congress occasionally makes noises about eliminating the fourth preference visa category. Given all these issues, filing the I-130 visa petition as soon as possible might be a good idea. Consult an experienced immigration attorney for a full analysis.