The actions/checkout issue 183 proposes a few options:
For instance (to be tested)
What do you think about being able to specify multiple ssh keys. For example:
ssh-key: |
 ${{ secrets.my_main_repo_deploy_key }}
 ${{ secrets.my_other_repo_deploy_key }}
I think if one doesnt work, it will fallback and try the next. To be clear, this currently won't work - would need to update the action to support it.
Or:
I already use a deploy key to pull in a python dependency from another private repository 'B' into the build of our repository 'A'.
For this to work, I set up a private key via a configured secret in 'A' and the respective public key in 'B' and use the following step:
     - name: Setup access via public/private key.
       # Below command requires the FOOBAR_PRIVATE_KEY to be configured via github repository secrets.
       # Also the key's public part must be added to the foobar repository deploy keys.
       # A private/public key pair without password (required in this case) can be generated with ssh-keygen.
       # This part is used for the git/foobar part in requirements.txt
       run: |
         mkdir ~/.ssh
         echo "${{ secrets.FOOBAR_PRIVATE_KEY }}" > ~/.ssh/id_rsa
         chmod 600 ~/.ssh/id_rsa
Also:
On GitHub the problem is, that we need one key for each submodule. AFAIK the idea in PR #190 will only allow one key for all submodules.
But we need multiple if we have multiple private submodules.
You have a full workaround in "Using private git submodules in GitHub CI" from Maximilian Ehlers.
The OP jon adds in the comments
I got it working: I wasn't putting the "-----BEGIN OPENSSH PRIVATE KEY-----" and the END in the secret; adding those fixed it.
Note: trying to replicate that on your workstation, using multipe SSH keys, will mean using ~/.ssh/config in order to reference those keys under different Host entries.
See "Enable Multiple SSH Key for GitHub on Windows 10" as an example.
Note: the issue 116 "private submodule checkout fails" now (July 2022) includes as an alternatives:
This solution works when you want to keep flexibility of URL repos and still use GitHub Actions with Deploy Keys to access private submodules:
  - name: Checkout
    uses: actions/checkout@v3
  - name: Clone Submodule
    run: |
        mkdir -p $HOME/.ssh
        echo '${{ secrets.SUBMODULE_REPO_DEPLOY_KEY }}' > $HOME/.ssh/ssh.key
        chmod 600 $HOME/.ssh/ssh.key
        export GIT_SSH_COMMAND="ssh -i $HOME/.ssh/ssh.key"
        git submodule set-url <path-to-submodule> git@github.com:<organization/submodule>.git
        git submodule update --init --recursive
        git submodule set-url <path-to-submodule> https://github.com/<organization/submodule>.git
        unset GIT_SSH_COMMAND