You don't really have to request an application access token. You can simply assemble one yourself.
An application access token is formatted like this:
app_id|app_secret
That's the application's id, a pipe character |
and the application secret.
Make sure that this app token is not accessible to your users! Only use and reference it on the serverside - never pass this to the client. For more info, check out the last few sentences in the relevant documentation.