What is Vietnam's view on things like LGBT issues?  

For the most part, no one collectively has targeted the LGBT people in Vietnam but they also do not go out of their way to protect them. They get treated differently in some areas but as a guest, this should not be anything to worry about. I believe in 2015, they made it legal to change your gender for trans' rights. Ho Chi Minh City is probably the most welcoming area and has a lively LGBT scene. 

It shouldn't be too much of an issue, really. So long as your cousin is polite and adheres to the social cues in the area, no one will care too much. He might get some looks from older folks but most people, in my experience, don't seem to care these days. 

There will be people who judge you no matter where you go but you are very unlikely to b physically targeted or harassed over it in Vietnam. People there mind their own business and don't want trouble.

I don't think he has anything to worry about so long as he is respectful in public. Holding hands, kissing, hugging, most people just look the other direction and no one seems to care but he should not dress lude or act provocatively in public as this will likely draw negative attention and offend people. 

Having gone with a gay couple (a friend of mine and his husband) to Vietnam, I have not noticed any issues. I think that Pho is right. So long as you are respectful in public, people just don't care all that much.