I suspect your son is suffering from anxiety -perhaps generalized anxiety or social anxiety. If anxiety is the issue, then I assure you that your son will not outgrow it nor will it go away. You were asking whether you should seek help - the time to intervene is when your son's anxiety is interfering in his life (and frankly, that is occurring now). Our child suffers from severe anxiety so we have been on this path for some time. Today, she is doing very well with proper and aggressive treatment.
Anxiety is a mental health condition which "has its origins in brain chemistry, not in real life experiences. That is why it persists in spite of the reality that your son has not experienced traumatic events in his life" - the parts in quotations were copied from Dr. Kennedy in the "ask a doctor" forum - child behavior section - titled 'severe anxiety in 5 year old child" posted on Dec 10, 2008 (if you would like to read Dr. Kennedy's reply).
First, I might suggest that you google words/phrases as "childhood anxiety" or "anxiety at school" or "generalized anxiety" or similar words/phrases to learn more about this condition. Then, since it appears that your son's anxiety is interfering with his life, it seems prudent to speak to your family physician for counsel re this issue. If he/she is unable to help you, then ask for a referral to a medical specialist with experience in anxiety issues. By the way, anxiety is very common in children and also highly treatable. I hope this helps ....
Thank you JDTM - your reply is helpful and makes a lot of sense - I will look at the sources you suggest and take the appropriate steps - it's reassuring in a way to know this is common though saddening to read they don't grow out of it.
I hope your daughter continues to do really well.