I think it depends on the type of Vietnamese. I've got loads of Vietnamese friends who are really Southern Chinese, but their families moved to Vietnam a hundred or so years ago. There are a lot of Thai people like that too (although the Thais I know with Chinese background a far less likely to think of themselves as Chinese). I think most Vietnamese people are probably mixed Chinese/Vietnamese. I've only met a hand-full of Vietnamese people who looked Malay (and I've been told - by Vietnamese people - that they are "real" Vietnamese. Generally I do consider Vietnamese people "Asian" though.
For me Asian means either having predominately Sinic facial features or living in a culture that it dominated by Chinese culture. So: China, Vietnam, Tibet, Mongolia, Korea and Japan. I consider Thailand, Cambodia, Laos and Burma to be their own thing because the indigenous populations look Malay, not Sinic, and also because they seem to be more heavily influenced culturally by India than China. The Philippines is kinda the outlier of all of it . . . generally I consider them Pacific Islanders though (Austronesians).
I've written this before... But I don't consider "South Asians" to be Asian. I also don't consider the Indian subcontinent to be part of Asia. For me "Asia" is China eastward. This is largely because South Asian culture and the way people look is so different. I actually tend to think that Indians have more in common culturally (and clearly physically) to Europeans than East Asians proper.