This will no doubt seem like a recklessly radical solution, but it's what I would do. The boxes at either end of the bellow in an accordion are commonly called "cajas armónicas" in Spanish. A caja armónica (or caja de resonancia) is a sound box or soundbox. Not a sound board; that's a tabla armónica. A piano has a sound board (though it's often called a caja armónica in Spanish); a violin has a sound box.

However, the wooden boxes at the ends of an accordion are not actually sound boxes, so it's not surprising that they're not called that in English. A sound box is a hollow space that amplifies and enhances the sound of the instrument. Some accordions have been made in which the boxes contain sound boxes (sometimes called tone chambers of tone boxes), but it's not actually usual, and if they are present they're just one element within the whole box. The boxes are simply housings containing the reed chambers and, as your text says, the keys or buttons that produce the notes by directing the air from the bellows to particular reeds.

In accordion jargon these boxes are simply known as the "ends": the treble end and the bass end.

So I would advise you not to use the literal translation, sound box, but simply to call them wooden boxes, and you could add "or ends".

"Parts of the accordion
The accordion consists of two wooden boxes joined by a bellows. These boxes house the reed blocks for the right and left hands. Each side has grilles in order to allow air in and out of the instrument, and to allow the sound to better project."
http://www.accordion.co.uk/accordion-faq.html

"The most important components of the accordion are divided into three main sections namely, the expandable bellows and two wooden units known as the treble and bass ends respectively."
http://www.carlinmusic.net/the-button-accordion/

A genuine sound box or tone box, present in some accordions, is known as a cassotto:

Cassotto / Tone Chamber
A tone chamber is a cavity designed into the internal structure of some piano and chromatic accordions through which the sound from selected sets of reeds (normally Bassoon for single cassotto / bassoon and clarinet for double cassotto) must pass before reaching the listener.
The main purpose of the tone chamber is to give those reeds a more mellow sound.
It also makes some improvement in the response time of the lower notes."
http://www.accordion.co.uk/accordion-faq.html
These accordions will also typically have hand made reeds.

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Note added at 57 mins (2015-04-17 12:19:58 GMT)
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Typo in paragraph 1: bellows, not bellow.

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Note added at 1 hr (2015-04-17 12:46:51 GMT)
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Or if you like, you could perhaps say "two wooden boxes, known as 'ends'", so as to use a more or less technical term.

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Note added at 4 hrs (2015-04-17 15:57:57 GMT)
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By the way, on "tabla armónica" there's this previous question, which I answered:
http://www.proz.com/kudoz/english_to_spanish/music/5544239-s...