The Austrian Olympic Committee (AOC) said Vanessa Sahinovic was "severely injured” with multiple fractures and has been flown to Vienna for further treatment.
15-year-old Luna Pajer was also flown back from Azerbaijan to the Austrian capital after suffering injuries to her arms. Verena Breit bruised her right thigh and has returned to the village.
The AOC said the collision happened at 8:30am local time on Thursday, when the young athletes were walking on the pavement in the Olympic village.
The Local has been sent a shocking video which apparently shows the accident - in which a bus is seen travelling at some speed, and ploughing into a group of people.
Warning: Graphic footage
Reporter Khayal Verdiyev told The Local that the Azerbaijani government has removed the video from local news websites.
AOC chief medical officer Dr Alfred Engel said Sahinovic had suffered a polytrauma and multiple fractures. Payer's arm injuries "need further medical clarification", he added.
AOC president Karl Stoss said in a statement that the accident was a "severe shock", and that it had been a very difficult day for everyone involved.
"The European Games had not even started, but already the Austrian team has suffered a heavy blow," he said. The team will be receiving counselling.
The inaugural European Games begin on June 12th and feature 20 sports, 16 of which will be included in next year's Olympics.
The government of Azerbaijan has banned a number of media outlets and human rights activists from entering the country to cover the games.
Azerbaijan has been heavily criticised by human rights organisations for imprisoning journalists and activists in the run up to the games.