I have gotten a bunch of hits lately from people searching for information about how they can adopt and or foster a dog from Japan. I have also gotten emails about how people can adopt animals from Japan and I know I am not the only one.
The answer is right now the Japanese government is not allowing dogs or cats to leave the country whose owners are still missing or who cannot claim their animals because they themselves are living in shelters. The Japanese government understands that it could still take weeks for owners to find the shelter their animals were taken to if they were found and then possibly months until the owners have a home or temporary place to stay where the animals are allowed to be with them. Some owners who are finding their pets in the various shelters around the country are giving permission for them to be placed in foster homes within Japan. The owners know that by giving permission for their pets to go into foster homes so they don't have to stay in the shelters that they may never see their pets again because according to Japanese law once the pet has been in this other home for a certain amount of time the foster home can legally keep the pets.
If you know someone in Japan with a pet whose home was destroyed and you are able to get in contact with them you could speak to them about the family with the pet coming to stay with you for a few weeks. Some Japansese families are sending their children to stay with family or close friends so they can get back to some sort of normal so many offer to care for the pet as well in that situation. Sending dogs or cats from Japan to the US might not be the easiest task but it is possible. Birds are a problem (if they were born in Japan they can't come to the US at all) and small animals like rabbits ferrets, etc are also a problem to fly from Japan to the US.
Right now the Japanese government has dropped the quarantine requirements so pets can leave with their families without having to wait the required time. Paperwork still has to be produced on both ends, in Japan and the country the animals are flying to. For the families who are leaving the cost of taking their pets with them is pretty high. It is determined by weight and size of crate just like anything else being shipped cargo. Unless they are accompanied by someone they cannot fly in cabin or as extra baggage.
For those who felt getting their children out of Japan was their first priority and that their pets were left somewhere safe are finding it to be cost prohibitive to get their dogs and cats out of Japan. The reason this is happening and I think it would happen in the case of taking dogs or cats from shelter and shipping them to foster / adoptive homes in other countries is because they are insisting that there be a customs savvy middle person hired to ensure all paperwork and vaccinations are in order so the animal is allowed to leave Japan and arrive in the new country without spending 6 months in government quarantine. I was told the cost of this service is about $3,000 US Dollars and would include pick up in Japan any trips to the vet for health certificates / vaccination (though actual vet work might not be covered) and hand delivery to the airport to get the pet on the flight.
I also wanted to point out that even if their owners did not survive the Tsunami family members from other parts of Japan are searching for the animals. I have heard of a lady who took in her brother's dog once it was determined he did not survive because she wanted something of his that he loved and cherished. I also read a story of the grandparents surviving and are with the children in a shelter but the parents are still missing where the grandparents once they have more stable housing would like to find the missing pet if possible for their grandchildren.
The best thing you can do right now is make a donation to either WordVets or JEARS to help care for these animals because they will be doing so for months to come.