Omer,
AddtoAny module doesn't have an API to change the links, so you would have to edit the code in the module. If you look at line 162, there is this:

if (is_object($node)) {
$link_name = rawurlencode($node->title);
$link_url = rawurlencode(url('node/'. $node->nid, array('absolute' => 1) ));
} else { // Front page
$link_name = rawurlencode(variable_get('page_title_front', variable_get('site_name', '')));
$link_url = rawurlencode(url('', array('absolute' => 1)));
}

If you change this code to the following, it should work:

global $user;
// Check if the user can act as an affiliate
$aff_query = (user_access('act as affiliate')) ? "?a=". $user->uid : '';

if (is_object($node)) {
$link_name = rawurlencode($node->title);
$link_url = rawurlencode(url('node/'. $node->nid, array('absolute' => 1, 'query' => $aff_query) ));
} else { // Front page
$link_name = rawurlencode(variable_get('page_title_front', variable_get('site_name', '')));
$link_url = rawurlencode(url('', array('absolute' => 1, 'query' => $aff_query)));
}

The code just checks if the user can be an affiliate, and if so, adds the aff string to the url.

Sincerely,
Leighton Whiting