It is possible to name an organization as a potential guardian for your pet in your pet trust. However, shelters cannot and do not guarantee that they would be able to find a new home or that the new home would be suitable.
There are some companies that offer a service for a fee or donation to care for your pet when you no longer can. Not all facilities of this type are reputable, however. Before choosing a location, visit it to see how the animals are cared for and socialized. If you find a facility you trust, reassess it regularly. Changes in ownership or funding may alter the level of care their animals receive. If the organization will assist in re-homing your pet, learn about their selection criteria to make sure you agree with where they will put your pet.
If possible, it is best to select a guardian you trust to oversee the process of finding your pet a new home.