They are doing it the correct way. In order to get union recognition you need to get enough people in the company to sign union cards. This is the normal way to create a union.

Companies argued that some people could be coerced and so the NLRB created a "two step" process in case the company protests the creation of the union. Basically if the company says "we don't think these cards were signed under proper circumstances" the NLRB would come in and hold a secret ballot.

That is basically what's happening here, and "wikipedia's side" is actually them reading a statement written by their lawyers in order to activate the NLRB balloting process.

The real reason why the companies like to invoke the secret ballot process is it takes a long time, and this gives them a lot of time to fight the union, fire the organizers, and "convince" employees to withdraw their support.