Where English uses 'don't/doesn't', Italian simply has non before the verb it negates. The verb still needs to be conjugated.
Ex.: mangio il riso - non mangio il riso
compriamo la cioccolata - non compriamo la cioccolata
'we DID not buy the white shoes' - Non abbiamo comprato le scarpe bianche
I am afraid that would not be correct.
le can be the article 'the' (in its feminine plural form) or the pronouns 'to her' or 'them' (plural feminine form).
The position in the sentence clarifies its role.
If le meant 'to her', then it would be placed before the verb: noi non le compriamo (le) scarpe bianche. This last sentence could also be read as 'We don't buy them, the white shoes'. It's redundant in English, but it's fine in Italian.
In this case, it follows the verb, therefore it is used as article.
If you are a beginner, there is no difference ;-).
If you want to speak proper Italian, then you need to keep in mind that using the personal pronouns as subjects (io, tu, lui,lei, noi, voi, essi, esse) in front of a verb, means that you are putting extra emphasis on the subject.
The difference between the two sentences is that the latter should actually be read as 'it is us who are not buying the white shoes"