One of the easiest ways to rescue a Yorkshire terrier is to find a rescue organization that specializes in that breed and apply to adopt from them. These rescues are often available through breed clubs and the American Kennel Club.
It is helpful to look for a Yorkshire terrier rescue in your general vicinity rather than one far away. Some rescues refuse to set up adoptions with people from other states. Finding a local Yorkie can also make it easier for you to meet the dog before the adoption and confirm you are a good fit for each other.
You may also be able to find a Yorkshire terrier in an all-breed rescue such as the city pound. Search pet adoption websites such as Petfinder.com to see if there are any available in your area. It is also a good idea to contact the shelter directly and ask them to let you know if they get a Yorkie that is available for adoption. Purebred dogs are often adopted quickly so this may help you get to the dog first.
Once you have found an available Yorkie, you can apply to adopt it. This usually involves an interview with shelter staff to make sure you are prepared to care for the dog, signing a contract and paying an adoption fee. Some rescues may have additional requirements, such as a home visit or trial period.