No, this does not work.

It is tricky, because the infinitive form found in the dictionary

man kauft sich etwas

uses sich, but this has to be replaced by the appropriate personal pronoun, matching the man -> wir transition, which in the example case requires the plural uns. Sich is clearly singular and would not match grammatically.

It is of course possible to buy something for somebody else, as in

Wir kaufen ihm/dir/mir etwas.

but as I understand this is not the scope of your question.

No, this does not work.

It is tricky, because the infinitive form found in the dictionary

sich etwas kaufen

uses sich, but this has to be replaced by the appropriate personal pronoun, matching the earlier occurrence, which in the example case requires the plural uns. Sich is either singular or third person plural and would not match grammatically.

The full list is: ich/mir, du/dir, er-sie-es/sich, wir/uns, ihr/euch, sie/sich.

It is of course possible to buy something for somebody else, as in

Wir kaufen ihm/dir/mir etwas.

but as I understand this is not the scope of your question.