How do you extract vitamin c from vegetables?

You can't because vitamin A doesn't exist in carrots. Beta-carotene on the other hand is a provitamin A molecule that can be extracted. You can either blend the carrot and then extract the beta-carotene with acetone or you can use something like ethyl lactate.

For more information there is an article

Ishida, B. and Chapman, M., "Carotenoid Extraction from Plants Using a Novel, Environmentally Friendly Solvent",J. Agric. Food Chem., 2009, 57 (3), pp 1051-1059