Can you buy a coat of arms?

Not the way most sellers suggest. A shop selling the "family coat of arms" for your surname is selling a picture of arms that once belonged to someone with that name, and buying it gives you no right to them. What you can pay for is an official grant from a heraldic authority, if you qualify.

In English and Scottish heraldry, arms pass from the person they were granted to down to that person's descendants. A print of the arms of some long-dead Miller does not make you his heir.

Official grants come from bodies such as the College of Arms in London, and the fees are set out in how much does a coat of arms cost. For a gift or a reunion, you can design new arms in the coat of arms maker at no cost.