It depends. When referring to a specific region, you should capitalize the words East Coast such as “I’m traveling to the East Coast” since “East Coast” is a proper noun in this case.
However, if you are referring to a general region, such as “the east coast of the United States” then you should keep the words lowercase.
Confusing, we know, but the capitalization rules for east coast are related to why directions are also sometimes capitalized and sometimes not. You can read more about the rules for directions here.