Sablikova, a five-times Olympic medallist in speed skating, was not included in the International Cycling Union's (UCI) final list for the Rio Games despite the Czech federation nominating her through its own qualification system.
She won the Czech national time trial title and finished ninth at the worlds in Richmond, Virginia, last year.
Czech officials said the root of the problem was the interpretation of the qualification procedures for Rio.
The UCI turned down the Czech federation's appeal and now her only hope rests with CAS.
"I have worked very hard over the past three years in preparation for these Games," she said.
"I believe that I still have a chance to make my dreams come true and be part of the Olympic Games in Rio. If there is a chance I want to fight for it. That is why I travelled to Rio and the support I have received has been amazing."