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If I say I am Vietnamese. Is Vietnamese a noun or an adjective? According to OALD, Vietnamese can be a noun or an adjective. If it’s a noun, why don’t we say I am a Vietnamese? Because its meaning is a person from Vietnam.

The ODO says

Vietnamese

ADJECTIVE

Relating to Vietnam, its people, or their language.
‘He liked Asian people, Vietnamese people in particular, and their culture, considerably more than he liked Australian culture.’
‘We were representing a Saturday morning Vietnamese language school.’

NOUN

1 A native or inhabitant of Vietnam, or a person of Vietnamese descent.
‘From the 1400s on, the Cambodians lost territory to both the Siamese and the Vietnamese.’

2 [mass noun] The language of Vietnam, spoken by about 60 million people. It probably belongs to the Mon-Khmer group, although much of its vocabulary is derived from Chinese.
‘Lady Borton, who speaks Vietnamese, finds the atmosphere far more hospitable now than in the early years of independence.’

In

I am Vietnamese.

Vietnamese is an adjective. This is the correct way to say that you are from Vietnam.

Vietnamese can be a noun in certain usages. There is the usage when referring to the language in NOUN 2, and there is the usage in NOUN 1,

The Vietnamese are known for their exquisite cuisine.

Notice the usage of the definite article. This is what the Cambridge Dictionary says on the matter:

the

• used before some adjectives to turn the adjectives into nouns that refer to people or things in general that can be described by the adjective:
  She lives in a special home for the elderly.
  The French were defeated at Waterloo in 1815.

There is also

^?I am a Vietnamese.

The Cambridge Dictionary suggests this usage is grammatically fine:

Nationalities, languages, countries and regions

from English Grammar Today

When we refer to a nation or region, we can use:

[...]

– a singular noun that we use for a person from the country or region: a Turk, a Japanese, a German, a Brazilian, an Asian

However, in my opinion, at best, I find this usage questionable (^?) or awkward. To me, it borders on ignorant or offensive, especially if it's said about the person (e.g. ^?She is a Vietnamese). That's why you're less likely to hear it. So although it is grammatical, my recommendation is to reword it (e.g. She is/I am Vietnamese).

Note that the issue is with the singular noun with the meaning in NOUN 1 above and that it is not unique to "Vietnamese". I did not intend to provide an exhaustive list, nor do I intend this to be a perfect rule, but my cursory observation is that the usage of the indefinite article with -ese, -ish, and -ic forms is awkward or incorrect (*): *a Vietnamese, *a Chinese, *a British, *an Arabic. There are also other cases, such as *a French and *a Dutch.

You can find a longer list of nationality adjectives here. The issues have also been discussed on ELU and ELL in the following:

• Can I say “A Chinese” in English?


• Why can we say 'an American' but not 'a British'?


• 
  Why is "a Japanese" offensive?.

According to Oxford learner's dictionaries about Vietnamese, Vietnamese is a noun and an adjective too.

1. Vietnamese is an adjective:

To answer a question like "What nationality are you?". You will answer this question:

I am Vietnamese.

In this case, it is used as an adjective like:

• I am American


• I am Italian


• I am French

To mention Vietnamese people, add "the" before "Vietnamese".There are some example:

The Vietnamese were very friendly and kind.

You can use "the Vietnamese" in following case:

I am the Vietnamese who sang the USA national anthem yesterday.

2. Vietnamese is a noun:

You can use "a Vietnamese" to show a sentiment (a pride or a shame)

I am a Vietnamese and I don't want anybody to criticize my country