First of all, the fact that a pediatrician is recommend Flintstones vitamins is kind of scary. Has he read the back of the bottle??
There are other things you can do to boost his immune system like giving him probiotics (especially if he has ever been on antibiotics).
I give my kids
Juice Plus+ and am so happy with the results that I became a distributor. The gummy version is really tasty but has no artificial dyes or other additives. This has been independently researched and *shown* to improve children's health. While he may eat lots of fruits and veggies, he's probably still not getting enough--especially since cooking veggies often destroys the nutrients.
You might also want to try juicing. That's another way to get lots of raw fruits and veggies into kids.

When my kids seem like they're getting sick, I up the Juice Plus and also give them some echinacea (Herbs for Kids).