The Azure CLI doesn't support file transfers directly; there's no az scp or az ssh scp command. But it does support OpenSSH-based applications indirectly.
First, run this command:
az ssh config --ip * --file ./azure-sshconfig --keys-destination-folder ./azure-keys
The az ssh config command will create a local azure-sshconfig file along with an azure-keys folder containing an OpenSSH-style configuration and keys that are capable of accessing the Azure VM. Note that the keys have a limited lifetime; as of this writing, they will expire after an hour, and that doesn't seem to be configurable.
Once you have your config and keys, you can use other OpenSSH utilities such as scp and sftp, both of which take a -F ./azure-sshconfig argument to instruct them to use your temporary keys:
scp -F ./azure-sshconfig