Happy15

Using Vietnam passport you don't need a visa to enter Vietnam!
 
Using a Green Card to Enter Vietnam you need a Vietnamese Visa to enter Vietnam...
 
If you a Green Cards Holder if you go longer then 6 Months  you need Apply A US Travel Document as know I-131 ..US travel document you can go more then 6 months enter bạck to US with No problem..

You can using Vietnam Passport to enter Vietnam, and using  Green Card to re-enter USA.. if you using Green Card to enter Vietnam you need to apply Vietnam Visa.. any Travel Agency can apply Vietnam visa for you.." if your wife go more then 6 months she need to apply USA Travel Documents as know I-131

use your case# and IIN# login to pay the which fee is available.. I think the AOS should be available pay now...since you already have the case# and IIN#
https://ceac.state.gov/IV/Login.aspx