Can she get permit to travel from the home office?
She already has that! It is called a 2-year spouse visa! Look at the visa ... it will say "Mult" .... multi-entry ... so she is allowed to leave the UK and to re-enter ... to resume her residence in the UK.
For Spain ... well Spain will say that she will need a Schengen visa in order to enter .... but expect posts to follow this saying that it is technically not necessary to get that Schengen visa. But getting it will cut out any possible problem checking-in at the UK on the journey to Spain.
Also if my wife wants to go back to Vietnam for a month or so in the winter. Will that affect her Visa status and ILR application when her 2 year visa expires?
No problem going abroad for that length of time. It will still be clear that she will be living in the UK.
Also bear in mind this. After 3 years in the UK she might want to apply for Naturalisation as British. In the 3 years up to the application date for that she should not have been out of the UK for more than 270 days, nor out of the UK for more than 90 days in the last year up to when she applies.
But clearly the sort of absences you are talking about will not cause a problem in respect of either of those tests.
