No, you do not have to have your vessel with you at all times.  Your vessel acts as a focal point in which you and your companion can connect with focused energies, but they are free to move around and are not bound to the vessel.  They are linked to it with flexible bindings, but they are no way imprisoned in it so you do not even have to have your vessel with you to connect with them.
Many people find the vessel helps them to connect with their companion due to the concentrated energies of the bindings so it can make it easier for connection but you do not have to wear it or have it on you at all times to interact and work with them.
If you would like to have your vessel with you, you can wear it, display it in an altar, keep it in your pocket or purse or carry it around with you through other means.  There is no right or wrong way to display your vessel.