My dog has been having issues with throwing up food?

The periodic vomiting isn't terribly normal. Some dogs may throw up due to low blood sugar, esp if you do not feed them regularly at least every 12 hours. Yes, it could be caused by feeding chunks of veggies. Try cutting up the carrots into slices or get a food grinder (Northern Tool) and puree the veggies. Dogs don't digest plant cellulose unless it has been one of the following: pureed, lightly steamed, previously frozen or fermented (like sauerkraut).