Regular druids I would suspect are omnivorous or vegetarian. The fighting types seem to like hanging out as bears and making friends with wolves so those guys are probably not strictly veg.
I would imagine they are pretty heavy into locally sources or hunted foods though, no way they would be buying a tube of pink meat from Wal-Mart.

Shadow druids are another story entirely, given their bond to plant life I would expect them to be almost entirely carnivorous, and likely man-eaters,
(I was going to use the word cannibals there but is a wood elf that eats you in bear form being a cannibal?)