I don't know if LSAC will grant an appeal, but I do think you need to consider whether there is any point in taking it again. If you've taken the test three times and have been disappointed with your score all three times, you may need to 1) accept that you're scoring within your abilities or 2) take some serious time off to prepare.
I would only consider this last option if I had some real, concrete reason to believe that I would score higher. For example, if I didn't have time to prepare, didn't take a prep class, etc. But if you properly prepared and put in the time, it's unlikely that your score is going to magically increase simply by taking the test again.
I think most people end up scoring lower on the LSAT than they hope. At some point, you just have to take what you've got and apply.
Just my two cents. Good luck with your studies.