The shelter is required by law to follow certain guidelines but this is what I would do.
1. Call the shelter and find out who the local Yorkie rescue person is/phone
2. Call this person and tell them your interest in this dog - no matter what the costs
3. This dog could easily be a biter - they get so frightened in the shelter & the shelter may only release to a rescue if they have a good working relationship otherwise they may have to put the dog down no matter what.
4. The rescue person could convince them the dog is very ill and please do an early release to them
5. Check your laws - in Arizona it is only 3 days if the dog does not have ID tags or a micro-chip - 7 days if it does.
6. Rescue could easily have a great relationship and work behind the scene with the shelter to get the dog released to them - something that would be very hard for you to do.
7. If rescue can't help you then find out what day and time the dog will be available for adoption and show up 10 minutes early. Usually small dogs have a waiting list and you have to be there at a certain time so everyone who wants the dog puts their name in a hat and whomever name is drawn is who gets the dog - doesn't mean they could be a good care giver
. If you win the drawing then immediately take it to a vet an get a full blood panel work up on the dog. It is better to know what battles you might have to face with this little one. You also do not want this dog around yours until you have the facts or diseases could be passed to your dog. I always have a 14 day separation period if the dog came from off the street. I just kept the new one in a separate bedroom. Please as much as you feel bad for this little one DO NOT put your dog and this dog together until your vet has seen this dog or you could be hurting your dog..Tough Love.
Good Luck and lets keep our fingers crossed for this little one.