Orient are seeking permission for a judicial review of the bidding process which saw the Hammers named as top choice to lease the ground in Stratford, east London.
The League One club argue that they were not allowed to make the case for sharing the venue, which lies a mile and a half from Orient's Brisbane Road ground.
Owner Barry Hearn said today: "The rules of the bidding process created by the London Legacy Development Corporation we do not believe provided for teaming, which is for all parties to share the stadium.
"It is our legal opinion that this is a fundamental flaw. We have gone to the High Court to have the decision struck out."
Hearn added: "We are doing everything we can to protect Leyton Orient Football Club, which is endangered by West Ham moving to the stadium."
The legacy corporation's board chose West Ham in December.
A spokesman said: "We have been notified that Leyton Orient have made the decision to issue proceedings for judicial review.
"Whilst this is disappointing, we believe that our processes have been robust, fair and transparent and that the challenge is misconceived."
Orient said it "would not have issued proceedings if it were not confident of success."
The Hammers were named as preferred bidder ahead of Orient, Burnley's University College of Football Business, which wanted to use the 60,000-seater stadium as a new campus, and a consortium seeking to turn it into a Formula One circuit.
But the deal collapsed in 2011 due to legal challenges from Spurs and Orient.