Needing to restore Elements 15 from a crashed computer. Do I need a license?

All you need is the install file and serial number. If you didn't keep them, that may have been an expensive mistake. (Of course, you need to keep them somewhere other than on your computer, because you're most likely to need them after a computer failure).

Also depends on which operating system you want to restore Elements 15 on.

If you're running Windows, you should be OK.

If you bought for a Mac, you won't be able to reinstall PSE15 on a modern Mac (Intel or M1) running Catalina, Big Sur, Monterey or Ventura.

Only if you have the installation file or disc and serial number.

If you registered it, it should be available from your Adobe account.