You can buy orange craft foam in most places that do craft supplies (art & hobby shop etc.)... it's fairly lightweight.
Or make a cone shape from lightweight card or sugar paper (do they call it that any more??) (again, available in orange), and crumple it a bit to give it an authentic carroty look.
You could also make a rough carrot shape from something like a scrunched up plastic bag, and wrap it with orange wool to give a slightly uneven ridged effect?
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