Do LinkdIn care about bulk spam? I've had to put every single one of my email addresses on their "do not mail" list due to their software's habit of sending invites to your entire address book, even in some cases where the member has explicitly selected the option not to give them access. Otherwise I get spammed every time a former student joins.
I assume the app also sends invites by email, so tests: can you send other emails (if not the problem is with your email service), have you ever sent a successful invite using LinkedIn (if so it's not that the app is mis-configured)?