Oh, I am sorry to read what is happening.
It sounds like your son is having panic attacks. They can happen at a young age. They are often hereditary, so it is no surprise your husband has them.
I have seen panic attacks, and have had one myself -- you feel like you are dying, literally, the anxiety is that strong. They are very scary and are associated with real physical symptoms like heart racing and flushing, and even sweating. When you are going through them, it is hard to say to yourself "OK, this is a panic attack, I am NOT dying..." You just feel like you are dying! I am describing my experience as an adult - so I can just imagine the terror a child must feel.
I suggest that you find a therapist -- because panic attacks carry a hereditary predilection, they are likely not going to go away for a while. In adult psychiatry, medications and group therapy are what is generally used, with the therapy aimed at identifying triggers for attacks and exercises to "guide"oneself through the attack when it happens. Talk to your doctor about finding someone who works well with kids and is someone you are comfortable with - also, think of combining different forms of therapy, like single therapy and group or play therapy, to deal with his overall anxiety. I think that will help a great deal.
One more thing, I should add that, if he hasn't already, a visit to the cardiologist or your pediatrician should be in order too -- your kid's palpitations may be coming from a heart condition (which in turn may be making him anxious) - so that must be ruled out before considering panic attacks.
And another thing -- I doubt that this is a vaccine related issue. Often, the 4-6 year shots are the first set of shot a child remembers -- many children develop a lot of anxiety over getting shots. The way I am reading your story is that the shots triggered an underlying disorder of panic attacks (assuming a heart condition is ruled out). Panic attacks, if the tendency to have them already exists (your husband has them, and he is the kid's bio dad, right?) ultimately get triggered by an anxiety-provoking experience. It could be shots, or a scary roller-coaster ride, or a scary life experience, like being in a car crash. So, you can also consider vaccines, but I do not think this is the answer. My opinion only!
OK, and one last thing (I promise! You must have entered your "So What Happened?" post while I was editing my response) I have seen dozens, if not hundreds, of children get 5 vaccines and more! Where I trained, the pediatric clinic ran a pediatric care program for adopted children, and we often got kids who never had a vaccine at 6-7 years old. We would get them "up-to-date" all at once! It sounds brutal, but it was how it was done...and I never saw any ill consequence, ever. I have seen children with anxiety around vaccines. This part is normal. To still have anxiety after the vaccines is not normal. But still, if you think this was from vaccines -- how do you explain that his father also has anxiety attacks? There were much fewer vaccines when he was a kid. I would not attribute this to shots.