Orange carrots are a very “modern” phenomenon in the history of carrots: the orange colour was developed only in the 1600s, in Holland. The Greeks and Romans wouldn’t recognize our carrots: to them, carrots were purple or white. (Learn more about carrots.)
You can home can heritage carrots — the multicoloured yellow, purple, red, etc., carrots. They will taste great.
The vibrant colours will fade to paler shades during canning, though.
If you don’t mind that, that’s fine. They do of course still need to be prepped (including peeling) and pressure canned as per USDA methods and processing times for regular carrots (or what we now call ‘regular carrots’, at least.)
You can also use heritage carrots in pickled items, such as pickled carrots, where their colours might survive better. Just use them in place of the carrots called for in your tested preserves recipes.
Summary: To can heritage carrots, follow the exact same directions as for canning carrots.