To prove damages, an individual would need to prove that they were put at a measurably worse disadvantage than the general AMCAS applicant. I see people post all the time about submitting in June/July/August, and not being certified until Nov/Dec/Jan. This means that although frustrating, it is not an individual issue, and that you probably could not prove damages. If you are reapplying, and have problems, having an attorney send a threatening letter may help, but this is not because AMCAS has any fear of losing a lawsuit. It's just that to mount a defense costs money (although if they are sued and win a dismissal, they could counter sue to get their defense costs back), and its cheaper to resolve your issue.
AMCAS doesn't have a contract with applicants to do something in a timely manner, so you have no breach of contract either.
I see posts that complain about lawyers frequently, many of them with incorrect information (like that they don't have expensive malpractice insurance, bureaucracy, or high overhead like us poor doctors (to be)). Suing AMCAS just contributes to the problem of too much unnecessary litigation. You are mad. You're depressed. You're totally stressed. I completely understand, and also feel that I got screwed by AMCAS. But that isn't real damage, and especially if you got in somewhere this application cycle, it's time to let your feeling go and move on with your life. If you are reapplying, I'm sorry you will have to deal with AMCAS again, and hope that this year is smoother for you.
In the big picture, it seems like blaming AMCAS for your dissapointment is like blaming URMs for there not being enough slots at your favorite school. The process is stressful and flawed. This is the function of there not being enough spots at medical schools to accept even half the qualified applicants. The AMA has a big role in keeping the number of slots available low, so maybe you want to direct some of that energy at a system that can be changed, and not at other applicants or brand new AMCAS employees.
AMCAS had a turn over rate of three weeks all year (told to me by two different people who answered the phone). Why? Because they got screamed at and dumped on by applicants all day, for stuff that wasn't in their control. Applicants who take out their frustration on the wrong people contributed significantly to the problem of incompetent workers at AMCAS. If you are not getting what you want, ask nicely but insistantly to talk to a supervisor, or an employee who has been there for a longer time. That will get you a lot further than getting mad at the customer service rep who answered the phone.
Okay, I'm done ranting.