I think carrots in juice is fine, but drinking regular glasses of carrot juice, even straight from a juicer, might be harmful. I think it's one of those veggies that when taken in moderate quantities is OK but in large doses can actually be harmful.
"Nitrates are a chemical found in water and soil, and they're a concern when it comes to feeding your baby. Babies who ingest too much can develop a type of anemia called methemoglobinemia.
Preparing formula with well water that's high in nitrates is the usual cause of the illness, but some vegetables can also contain nitrates. The most likely sources are beets, carrots, green beans, spinach, and squash."
Source
ETA: I don't see carrots on the dirty dozen, nor on the clean fifteen, so who knows....