Hi
I think my first step would be to look at the regulations for where you are moving to. Countries such as Australia,New Zealand do not allow animals into the country from Vietnam. The only way around it is to quarantine them in somewhere such as Singapore for 3 months.
This at least was the case about 4 years ago when I looked at it.
If you are moving within Asia this is not a problem.
I would approach some of the major removal companies as I am sure 1 of them does this service as part of moving households.
An update on Hanoikiwi's info -
When departing from Vietnam, pets now have to be quarantined for a minimum of SIX Months !!!!! in a country - such as Singapore - before going to Australia. The pets are then quarantined for a further 30 days at least. They have limited space and bookings must be made.
(This helped me decide whether to adopt a pet here in Hanoi!)
The website of the Customs Department of the country you are heading to should be provide a fair bit of information.
