The Swiss club say they will make an initial claim of more than €4million (£3.47million), with reports in the Swiss press suggesting the final demand could be for tens of millions of Swiss francs.
Sion have also threatened to make a criminal complaint against UEFA after claiming a Swiss court backed their fight to be reinstated in the competition.
They were expelled from the tournament on September 2 for fielding ineligible players and saw their appeal against the decision rejected by UEFA's appeals body Wednesday.
Celtic, who lost 3-1 on aggregate to Sion in the play-offs, were handed their opponents' place in the group stages and travelled to Spain this morning for their opening match against Atletico Madrid which takes place on Thursday night.
A Sion spokesman told Press Association Sport in an email: "We send today a first order to pay to UEFA, and directly to its president Michel Platini, for more than 4million euros.
"UEFA believes that it rules the world. The law of the jungle couldn't rule football."
The spokesman said: "Yesterday [Wednesday], it was a clear case of misfeasance. Swiss Court won't let the case fall, because they obliged UEFA to reintegrate immediately Sion in the group stage.
"UEFA now infringes Swiss laws, that means that penal proceedings can be prosecuted."
Sion have now scrapped plans to travel to Madrid, despite the club's president Christian Constantin previously claiming his side would make the trip regardless of the UEFA ruling.
However, they believe they could still be reinstated at a later date and plan to take their case to the Court of Arbitration for Sport.
The spokesman added: "We're not flying to Madrid today. No [sic] for the moment.
"The players and coaches are very disappointed. But we still believe we can integrate the group stage later. We now wait for damages for the harm.
"An appeal to CAS will probably be made this week. We wait for the written judgment and explanations."
Reflecting on the outcome of Tuesday's appeal hearing, Constantin told Swiss newspaper Le Matin: "What UEFA have done is to ignore the law of a state and its institutions.
"By violating the decision of a judge, those responsible have aggravated their case.
"For these gentlemen, the real trouble is going to start now.
"It would astonish me too if the justice of this country allows itself to be trampled by an institution subject to Swiss law."
He added: "We must make up for the financial prejudice without delay."
February 15, 2008: Egypt goalkeeper Essam El Hadary signs a contract with Sion despite still being under contract with Al Ahly.
June 12: Al Ahly file claims with FIFA against El Hadary for breach of contract and Sion for inducement to breach of contract.
April 14, 2009: FIFA refers the case to its dispute resolution chamber.
April 16: The DRC awards damages of 900,000 euros to Al Ahly and imposes a ban on Sion registering new signings for the next two transfer windows. The decision is not communicated until May 29.
June 1, 2010: A Court of Arbitration for Sport appeal hearing upholds Sion's transfer ban, which had been suspended for the 2009-10 season. The ban applies to the January 2011 and summer 2011 transfer windows.
August 13, 2011: Pascal Feindouno, Mario Mutsch and Gabri all feature in Sion's 1-1 draw with Lucerne in the Super League, allegedly in breach of the FIFA embargo, after a Swiss court ruled in the club's favour.
August 17: Coach Laurent Roussey vows to select from his full squad - including the trio and two other players affected by the issue, Jose Goncalves and Billy Ketkeophomphone - for the first leg against Celtic.
August 18: Goncalves and Feindouno start in the goalless first leg, with Mutsch and Gabri featuring from the bench and Ketkeophomphone an unused substitute.
August 25: Feindouno scores twice as Sion win the second leg 3-1, with Mutsch also starting the game.
September 2: Sion are excluded from the Europa League by UEFA, with Celtic instated in their place for the group stages.
September 13: UEFA rejects Sion's appeal against their expulsion, dismissing a hearing from the district court of Vaud in Switzerland, where UEFA have their headquarters.
September 14: Sion announce they will take action against UEFA, looking for initial compensation of over four million euros (£3.47million), and potentially tens of millions in the long term. A club spokesman says: "UEFA believes that it rules the world."