Okay, so I get attacked with this alot, so I can answer this.
“East Asia” is a geographical term, not a cultural term. Therefore, Vietnam isn't part of East Asia, but is part of Southeast Asia, another geographical term.
If you mention East Asian culture, people will include Mongolian culture because that country is part of East Asia. However, its culture isn't related (mostly) to majority of East Asia, but is closer to Central Asian culture.
The proper term has to be Sinosphere: cultures within Chinese influence. Because that's what it is: Japan, Korea, Vietnam, and China are Sinosphere countries.
Why people hate it when Vietnamese are referred as East Asian? Simply because Vietnamese aren't East Asians (unless you count Jing ethnic group in China), but Southeast Asians within Sinospheric cultural sphere.
Also, they hate it because they're racist. Tons of them think Vietnamese are beneath them because they have darker skins and are underdeveloped countries. And they think that's okay, which is quite problematic.
So if you're going to group Vietnamese culture with China, Japan, and Korea, use the correct term so they won't have anything to grab you with. Use Sinosphere.