I dont really see an error, unless it is the space here:
$result = mysql_query("SELECT * FROM company WHERE Company_id = '$modify'");
//should be
$result = mysql_query("SELECT * FROM company WHERE Company_i ='".$modify."'");
If you are getting blank inserts, it's probably becuase the variables aren't being read due to the double quotes.
Should be:
$Query = mysql_query("UPDATE company SET company_name='".$ModifyCompany."' WHERE user_id='".$ModifyID."'");
edit
$modify could also be empty, check that like this:
if(!empty($_GET['id'])){
$modify = $_GET['id'];
} else {
echo 'Could not get ID.'; exit;
}