It's not wrong grammatically, but in English the word "attached" specifically refers to files that are sent along with e-mails such as documents or photos, whereas "included" would be best if the application were included as part of the body of your e-mail. "Added" sounds like you had previously sent the e-mail without the application, and are now adding it as an afterthought.
Option 1: I've attached the application form to this e-mail. (It is included as a Word/PDF document) Option 2: I've included the application form below. (It is just below the text you're reading now)