Is it possible to get the private key out of Azure Key Vault Keys?

Sorry, no.

Azure Key Vault does not support EXPORT operations: once a key is provisioned in the system it cannot be extracted or its key material modified.

You can do BACKUP, but all that's good for is a restore to Azure. You can't use it anywhere else.

As TerryCarmen said, you could not decrypt in local,only public key is available to the system. The API call to GetKeyAsync doesn't return private key data.This is why the DecryptAsync wrapper method does use the Key Vault API for descryption.

In other words, private keys never leave the vault, which is one reason to use Key Vault for decryption instead of bringing private keys in to the process.

For more details, you could refer to this article.