I think "Tôi không có tên họ" is a better translation.
I think they are equally good. Vietnamese two syllable words have often one syllable version. Like "vui vẻ" => "vui".
Colloquailly people prefers one syllable version because they're shorter and the media tends to prefer the longer version.
That's true, but I'm pretty sure 'tên họ' is the correct translation of family/last name. I'm not 100% because I'm a Việt kiều but I don't think 'họ' by itself, refers to family/last name. I myself have only ever used 'tên họ'.
Burmese don't have family name
Doesn't this also mean, 'I don't have them'?
That construction is awkward in Vietnamese for that meaning. You're much more likely to hear "They're not here", "I don't see them anywhere", "I could not catch them", or "they have escaped".
Some children don't have parents (because they don't know who their parents are) => they say: "I don't have family names."
I put in "Tôi không có một họ tên".
Why no 'một'? Why is "tên" not used?
As we are not specifying a concrete surname, một is not necessary (I dont know if it would be correct).
In English we could use also "I do not have family name". I believe that is correct (I'm not a native speaker)
In English, you still need the article for family name. Usually "a", but the could work if there's a specific name that's important in this context.