No, que que cannot appear in regular text except if the second que is a substantive or to mark hesitation/sluttering:

On ne doit pas utiliser que « que » pour traduire what.

Tu dis que… que j’ai tort ?

I would rephrase your sentence that way:

La seule chose qu’ils veulent, c’est que j’achète ce qu’ils vendent.

Or another valid substitute:

Ils veulent seulement que j’achète ce qu’ils vendent

Note that:

Ils veulent que je n’achète que ce qu’ils vendent

would fix the "que que" issue but is an incorrect translation has a different meaning.

The first translation says the only thing they have in mind is that you buy their things (but your are free to buy other things elsewhere too) while the second one says, one of the things they have is mind is that you buy their things and nothing else.

Ils ne veulent que que j’achète ce qu’ils vendent.

You have all the words needed, but not in the right order.

Ils veulent que je n’achète que ce qu’ils vendent.

Your English version only could use some rewriting.

They want me to buy only what they sell.

• They want [that] – Ils veulent que
• me – je
• to buy only – n’achète que
• what they sell – ce qu’ils vendent.