1. My son and I are U.S. citizens, but my wife is Vietnamese (she does have an American green card, though). Will this be a problem with resident permits?
There seem to be no issues with nationality when getting residency in Vanuatu. There are criteria you need to meet financially though.
2. Given my background teaching and writing and my wife's business experience, will we have a hard time finding work?
There is a website called Wokikik, it lists most of the available jobs in Vanuatu. Browse through these and put in some apps before you move to gauge interest in your particular skills?
3. I know that fruits and vegetables grow easily there, but that it hardly matters because these items are inexpensive in the local markets. But what about meat and dairy? We don't eat out a lot, but we do cook.
One of the things I struggled with in Vanuatu was the lack of variety with fresh produce. The fruit and veg at the markets is quite cheap when it is in season but after a while the same old bananas, coconuts, sweet potato and beans pall a little :p
Growing is seasonal and the seasons can be quite short over there for most fruits. It depends where you are on the island to what you will be able to grow in your own garden.
Imported fruit and veg are very expensive, even frozen varieties.
Beef is very cheap compared to Australia (most cuts around the 1200vt per kilo) but I don't know how it will compare to the US.
Lamb, pork and chicken are quite expensive.
Dairy is one of the most expensive things to buy there as it is imported and subject to import taxes. No fresh milk (uht only at about 250vt per litre), specialty cheese like brie, blue or feta average 5000vt a kilo. Even a basic cheddar will be about 2000vt per kilo. Sour cream 1000vt for a large tub. Whipping cream (uht) 1000vt a litre. Yoghurt 300vt for a small tub.
I have not used the schools though I have many friends with children that are happy with the schools (there are both French and English schools and fees are similar to Australian private school education). The general consensus is that the schools are great for younger children but many expats send their children to Australia or New Zealand for boarding school in the later years.
Someone else might like to answer regarding the hospital. I did not need medical care in my years there but when I first asked about doctors and hospitals I was told "feel the pain, get the plane"