A spokesman said the documents had been handed to League chairman Greg Clarke at the weekend.
Orient chairman Barry Hearn claims West Ham - who have also confirmed their interest in being tenants at the Stratford site - should be excluded from contention under Football League rules.
The League's regulation 13.7.5 states that consent to move to another ground shall not be granted unless it is reasonably satisfied that such consent "would not adversely affect clubs having their registered grounds in the immediate vicinity of the proposed location".
Hearn argues that his club would be badly affected if the Hammers were to move to the site, but West Ham insist they are "the right choice" to be the future tenants.
"West Ham United continue to press ahead with our legacy proposals for the Olympic Stadium," a statement from the Championship club said on Tuesday night.
"The club remain committed to east London and believe we are the right choice. In contrast, back in May 2009, our neighbours Leyton Orient were looking at a move away to Harlow or Basildon.
"The Olympic Stadium is in Newham and Leyton Orient is in Waltham Forest. We will not move away from Newham, after all it's our home borough.
"We have always focused on our legacy plan for the stadium, and in turn the East End and London as a whole, and already have the right permissions from the game's governing bodies to tender again."
West Ham applied for permission through the Premier League in the original tender process before their relegation.
A deal with West Ham and Newham Council for the Hammers to move into the stadium after the 2012 Games collapsed amid legal challenges, with the Government announcing last week that the stadium would remain in public ownership and be leased out.
Hearn wants to reduce the capacity of the stadium to 25,000 and is keen to discuss ways to "dig down or build above" the athletics track to make it more suitable for football use.