Resume or Application
What do they tell you to do? One of their sneaky little tests is to see how well you follow directions. If they say just to send a resume, just do that. Sometimes that might be enough for them to send you an application. That's a positive sign if that happens because it shows that they're interested enough in your quals to merit a formal application.
Absent any specific guidance, I might begin by sending a resume and cover letter, and request an app in the cover letter. Include a self-addressed, stamped envelope. I dunno how many pages pilot apps are these days, but you might consider sending a 9 x 12 envelope. Maybe you can download the form.
I would add that if someone you know at the company brings you an app that you should definitely get on the stick and prepare it and send it in, hopefully along with that someone's LOR. That is definitely a good sign.
Hope that helps. Best of luck with your job search.