Harry Potter and the Prisoner of Azkaban

Recently, I've decided to reread my favorite series of all time - Harry Potter. I've almost completed the third book and realized something. As many of you already know from reading the books, Sirius Black sent Harry a brand new Firebolt broomstick for Christmas. In my reading, I realized that Black couldn't have just walked into the quidditch supply shop and purchase a new broom and he couldn't have stolen it - it would have been obvious where the very expensive and stolen merchandise had ended up. So how the heck did Sirius purchase and send such a large object to Harry without getting caught AND anyone knowing who sent it or where it came from? (Also, he couldn't have purchased it as a dog... who would sell a brand new broom to a dog?)