WASHINGTON (AP) _ Air traffic controllers trying to organize a new union have submitted enough sign-up cards for the government to order a representation election, officials at the Federal Labor Relations Authority said today.

''We've checked them all over and it looks like it's adequate,'' Jessie Reuben, director of the authority's regional office, said of the sign-up cards and organizing campaign by the National Air Traffic Controllers Association.

The controllers submitted more than 5,800 union sign-up cards in early January to the labor relations authority. Under federal law, 30 pecent of the approximately 13,000 controllers employed by the Federal Aviation Administration have to sign the cards for the government to hold an election.

Reuben said the FAA is now putting up notices at all of its locations nationwide advising its employees that a sufficient showing of support has been made to hold the election.

He said the mail balloting could be conducted as early as April or May ''if there are no problems or hearings.''

Under the law, a majority of the working controllers would have to vote in favor of the union for it to become their collective bargaining agent with the FAA.

The controllers have been without a union since President Reagan fired 11,400 controllers in August 1981 after they refused a presidential order to return to their jobs.

Those controllers, members of the now-dissolved Professional Air Traffic Controllers Organization, went on strike in a protest primarily over working conditions. Since then, Reagan has refused to rehire any of the controllers who went out on strike.

The new union, the National Air Traffic Controllers Association, was launched with the help of several AFL-CIO unions. It is affiliated with the 22,000-member Marine Engineers Beneficial Association.