Yes, you should absolutely send a cover letter. Why? Because today, so many other people aren’t.

It makes you stand out.

It shows you’re professional.

It shows you’re willing to go the extra mile.

That’s what customer service is about, being willing to go the extra mile for customers.

If somebody is not willing to do that for their own career, a hiring manager is going to assume that they are not going to do it for customers.

So, yes, absolutely, for sure… Send a cover letter for any customer service job you’re applying for.