The Social Security Administration doesn't even mention submitting by mail as an option on its website, but for replacement ...

You can fill out the free Social Security card application form (SS-5) on the SSA website and submit it in person or by mail. Show valid, unexpired documents such as a U.S. passport, driver’s ...

Here's how to get a new Social Security card: Find out if you qualify to request a new Social Security card online. Create or use your my Social Security account. Fill out a paper application.