Travelling is definitely NOT required for most chef positions. Some corporate chef roles like at hotels or large chain restaurants might have a little travel involved, but not all the time. Depending on the role it can be helpful to get out and see different foods in the world as part of the creative process though.
But its not necessary to travel if you want to become a chef, but your work will fund your travel adventures, through working as a chef in a restaurant , hotel or a kitchen chain you will meet new people and give you the opportunity to cook in some unique locations.
Hope it helps, and good luck!