You can trade items for some Carrot Seeds. This guide on Where To Plant The Carrot Seeds In Pokemon Sword & Shield Crown Tundra will tell you where to find both the snowy field to plant the carrots and the field near the graveyard.
When you first enter the new town in Crown Tundra there is an old man standing by some fields. If you speak with him he will offer you some Carrot Seeds in exchange for some new ore that you can find in the Dynamax Adventure Dungeon. If you trade him the ore for the Carrot Seeds, he will inform you that you are not allowed to plant the seeds in the town. He tells you that you must find a new location to plant the seeds.
Where To Plant The Carrot Seeds In Pokemon Sword & Shield Crown Tundra
You cannot actually plant the Carrot Seeds straight away, if you traded as soon as you arrived in the village. Although you can obtain the seeds early, it’s actually part of the main story for the Crown Tundra DLC. As such, you will want to progress with Legendary Clue #1 before you can actually plant the seeds.
Once you progress far enough into the Legendary Clue #1 objective, you will be told by Calyrex that you can choose where to plant the Carrot Seeds. You can choose the field by the graveyard or the field covered in snow.
They are both marked on your map. This is a very important choice as you are choosing whether you get Calyrex Psychic/Ghost or Calyrex Psychic/Ice with the Iceroot and Shaderoot Carrots.