Miss Norkett, 20, from Llandarcy, Neath, had been the youngest player in the 2014 Rugby World Cup.
She died in a two-car collision on the A4109 Inter Valley Road, between Banwen and Glynneath, at about 19:40 GMT on Saturday.
The Welsh Rugby Union (WRU) said it was "deeply saddened to hear the tragic news".
Miss Norkett was in her final year of a sport development degree at Cardiff Metropolitan University and planned to pursue a career in rugby coaching.
Her family described her as "kind-hearted and caring".
In a statement, they said: "She was the youngest player in the Rugby World Cup and had gained four Welsh senior caps at the age of 17.
"Elli also represented Wales at Sevens and was selected for the Great Britain Students Sevens.
"On the club front she was proud to represent Swansea Ladies and the Ospreys.
"Elli was loved and valued by many and has touched the hearts of all her family and friends due to her kind nature and endearing personality."
A spokesman for the WRU said: "A talented, dedicated player and popular teammate, Elli was a treasured member of the Welsh rugby family.
"Our thoughts are with her family and friends at this difficult time."
A statement from the Ospreys said Miss Norkett had "her whole life in front of her".
"Not only was she a talented rugby player she was a good human being, probably the most important thing about her," a spokesman said.
"As an Ospreys in the Community coach, she was an inspiration to boys and girls of all ages who enjoyed being instructed by an Osprey and a Welsh international."
South Wales Police is continuing to investigate the collision, which involved Miss Norkett's silver Toyota Yaris and a silver Vauxhall Insignia.
A 42-year-old woman and her 12-year-old passenger sustained non-life threatening injuries in the crash.
A woman who died in a crash in Neath Port Talbot has been named as Wales rugby international Elli Norkett.