How have Scarface Tony Montana's mother and sister got green-cards and have been living in the US for years but Tony hasn't?
Context and dialog imply that back in Cuba, Tony was in prison for crime, while his mother and sister were law-abiding citizens -- but were not in the good graces of the Castro establishment because of Tony's late father's anti-Castro work. (For example, when first seeing him, Tony's mother snidely says to him "No postcards from jail?" Also, Tony mentions his dad's anti-Castro work.)
Then came the Mariel boatlift (depicted in the documentary-style short at the start of Scarface) when Castro opened the border briefly, allowing Cubans to cross the sea into Cuba. Tony's mother and sister seized the opportunity to escape, as did Tony -- though Tony left separately, from jail.
After arriving in the US (separately from Tony) his mother and sister were processed by the US government, identified as anti-Castro and therefore welcomed to the US as legals residents and eventual citizens. Tony, however, was identified as a criminal (a US agent examines Tony's tattoo and says "pitchfork means an assassin") so was stuck in a refugee camp. Time passed, but soon enough, Tony received and accepted Frank Lopez's request to kill Ribenga, and then Lopez was able to get Tony released from the camp.
Several months, at least, passed between Tony's release from refugee camp and Tony's visit to his mother and sister (first Tony works at the taco shop, then establishes himself with Frank Lopez). During that time, Tony's mother and sister had begun their lives in the United States, having known nothing about his whereabouts.
I can also state with some experience (my wife is Dutch, and when we first got married she applied for a Green Card) that it takes a LONG time to get processed for a Green Card. It's entirely plausible that Tony's mother and sister were selected for processing via a lottery system (something they still use today) and that Tony was simply left to wait his turn with everyone else.