1. General names
Chinese: The character 雀 is traditionally identified as meaning 'sparrow' but is more appropriately identified with the finches, hence traditional name of the family, 雀科 . (In fact, 雀 is used in some bird names, such as that of the peacock, that bear no resemblance at all to sparrows or finches).
A few traditional names of the finches are as follows (a full list of taxonomically systematised names can be found in the notes):
燕雀 'swallow-finch' Brambling Also 花雀 'flowery finch' or 花鸡 'flowery fowl' 金翅 'gold-wing' Greenfinch Also 金翅雀 'gold-winged finch 黄雀 'yellow finch' Siskin In Chinese, traditionally also refers to the Sparrow 朱顶雀 'vermilion-capped finch' Redpoll (or Twite, Linnet) 朱雀 'vermilion finch' Rosefinch 松雀 'pine-finch' Grosbeak 交嘴雀 'cross-billed finch' Crossbill Also popularly 交嘴 'cross-bill' 锡嘴雀 'tin-billed finch' Hawfinch
Japanese: Japanese is rich in names for the finches:
アトリ atori Brambling (written 花鶏 'flower fowl' or 猲子鳥 in Chinese characters, both etymologically unrelated to the word atori). ヒワ hiwa members of the genus Carduelis (written 鶸 in Chinese characters. The character 鶸 originally referred to a large kind of fowl in Chinese; its application to the finches is a Japanese innovation). マシコ mashiko Rose finches etc. (マシコ, written 猿子 'monkey' in Chinese characters, means 'monkey'. The name was extended to the birds on the basis of the monkey's a red face.) ウソ uso Bullfinch (written 鷽 in Chinese characters. The character 鷽 in Chinese was used for a bird resembling the Japanese paradise flycatcher (Terpsiphone atrocaudata) or Dollarbird (Eurystomas orientalis). Its application to the bullfinches is a Japanese innovation). シメ shime Hawfinch (written 鴲 or蝋嘴 in Chinese characters. 鴲 appears to be a Japanese creation; 蝋嘴 is the Chinese name of the bird). イカル ikaru Grosbeak/hawfinch (written 桑鳲 or in Chinese characters. 桑鳲 is a Chinese name for the birds. The character 鵤 was created by the Japanese based on the bird's 'angular' or kakubatta bill; the right part of the character, 角 kaku, means 'angle'). イスカ isuka 
Crossbill (written 交喙 'crossed bill' in Chinese characters. 交喙 is a Chinese name for the crossbills).
Vietnamese:
Vietnam does not have a large number of finch species and there is no particular name specifically referring to the finches. Most of them, in fact, are identified as some kind of Sẻ or 'sparrow'. Under Vo and Nguyen's original scheme, names are deployed as follows:
The Black-headed Greenfinch and Vietnamese Greenfinch (genus Carduelis) are called Sẻ thông 'pine sparrows'.
The Brambling (genus Fringilla) is treated as a Sẻ dồng 'field sparrow', which is what the Buntings are called.
The rosefinches (Carpodacus) are known as Sẻ hồng 'rosy sparrows'.The Brown Bullfinch (genus Pyrrhula) could be mistaken for a kind of Sẻ 'sparrow'.
The Scarlet Finch and two kinds of grosbeak (Eophona and Mycerobas -- a later addition) are known as Mỏ to 'large bills' (which may be related to the English term 'grosbeak').
Cross-bills are called Mỏ chéo 'oblique/slanting bill' .
In what appears to be an attempt to make the names more systematic, Nguyen, Le & Phillipps extend Sẻ thông 'pine sparrow' to four more species: the Brambling, the Eurasian Siskin, the Common Rosefinch, and the Brown Bullfinch. However, the result of this is confusion in the naming of the Brambling and the Black-headed Greenfinch.
One general V-E dictionary gives Chim sẻ khướư 'babbler sparrow bird' as the word for 'chaffinch', although these birds do not occur in Vietnam.
2. Species names
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