Pets are a part of your family and there is no reason that they should be turned away when seeking a rental home. Many pets end up homeless because of the difficulties finding a rental home that accepts pets.
There are many things to consider when renting a home with pets.
Just as you pay a security deposit, you will need to pay a pet deposit as well. This goes to the homeowner/property manager to ensure that any damage done by your pet(s) will be paid for.
Depending upon your specific rental, there may be a limited number of pets you can have, limited species – such as commonly aggressive dog breeds, or limited size for a smaller rental. These limits are implemented to protect the rental as well as other residents nearby.
By limiting the pets allowed, it also limits the tenant pool, which is something a homeowner/property manager will have to decide if it is worth it.
Limiting the pet specifics is also not a proven method of protecting the rental from pet damage. It is understood that it really depends on the pet owner. This is where references are a good indication as to how well the owner cares for their pets.
With acknowledgement of possible pet limits and good references, your family should be able to stick together through the move.