The Social Security Administration doesn't even mention submitting by mail as an option on its website, but for replacement cards with no changes, you should be able to fill out the Form SS-5 and ...

you may not be able to get a new Social Security card online. Residents of Delaware and Wisconsin can use the online card replacement service if they have a driver's license, but not other forms ...

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 ...