Does every last name have a coat of arms?

No. A coat of arms belongs to a person, not to a surname, so no last name has one by itself. Many surnames were borne by an armiger at some point, but most families have no ancestor who was granted arms. Sharing a name with someone who had arms does not make those arms yours.

In England and Scotland, arms pass down a family line from the person they were first granted to. Younger branches were often told apart with small marks of cadency. Two unrelated families named Taylor may each have had completely different arms, or none at all.

Customs differ by country. In Poland, for example, one coat of arms was often shared by many families of the same clan. The family crest vs coat of arms guide explains more, and how to find my family coat of arms covers the records.