In Jain dharma, carrots are root vegetables (kand-mool), so:

• Ideally: No, you should not eat carrots if you are trying to follow proper Jain dietary principles.
• This applies in both Digambar and Shwetambar traditions.
• The reason: when a root is taken out, the whole plant dies and many small living beings in the soil and inside the root are harmed. This is considered more himsa (violence), so Jains avoid potatoes, onions, garlic, carrots, beet, radish, etc. together.

If you are just starting and currently eat carrots, many people gradually give up root vegetables step by step, but the Jain ideal is to avoid them.

You can read more about why root vegetables are avoided here