. Since then it has been 4 months and the paper is still 'awaiting reviewer invitation'.
I find "awaiting reviewer invitation" to be a slightly strange phrase, but taken at face value it seems to mean that they still have not even asked anyone to referee the paper, let alone found a referee, let alone gotten the report back from a referee! If true that is indeed problematic: six months is waaaaaay too long to wait for this.
I have two plausible guesses as to what has happened to you.
1) They are having a lot of trouble finding a suitable referee. They tried on their own, failed, then asked you for help after two months (kind of a long time, but not a priori unreasonable). The people that you suggested somehow didn't pan out, and they reacted badly to that, putting your work on some sort of long-term queue.
2) Whatever electronic system is conveying to you the message "awaiting reviewer invitation" is simply malfunctioning. Possibly (as @RoboKaren suggests) the journal's email system is on the fritz as well. If that's the case, who knows what's actually happening: maybe your paper has gotten lost entirely. (Almost anything can happen. Once, almost ten years ago, when I inquired after a few months on the status of my paper, the answer was "We're having trouble finding any record of your submission. Are you sure you submitted to our journal?" This may be hard to believe, but...eventually we sorted it out -- in particular, they found the record of my submission! -- and the paper was published by that journal.) Or maybe the referee report is actually in and it is just the part of the system where they tell you this that is malfunctioning. (That also happened to me, much more recently, and again the paper was eventually published, although this time I am less thrilled with the journal and will probably not submit there again.)
Either way, you need to actually get in contact with an editor of the journal and find out what happened. Not responding to your emails at all within two weeks is certainly unacceptable behavior: every professional deserves to be treated better than this. In my opinion this total lack of response empowers you to take more vigorous steps to contact the editors of the journal. Such things include, in roughly increasing order of vigorousness:
A. Emailing the editors at their university accounts rather than the journal account.
(This is definitely the least intrusive: try this first.)
B. Sending multiple emails at the rate of one a day.
C. Calling the editors at any phone number they list on a professional webpage.
D. Enlisting a more famous / senior / powerful ally to try to contact the editor on your behalf.
E. Contacting other professionals to tell them that you must get in touch with Editor X and can't, e.g. the secretary of Editor X's home department, or the head of Editor X's home department.
I would not: contact an editor on a social media site like facebook, try to look up their home phone number or address, or anything like that. I think that's too much.
Once you do hear from an editor, find out how far off the rails your train actually is. After losing contact with you completely and advertising your paper as "awaiting reviewer invitation" after six months, in my view the editors should be apologizing and telling you what they're going to do to fix the situation. If you get the sense that they don't see it that way, it's definitely time to try again with a different journal.
