I have created a private Ethereum chain (via a genesis.json file, etc.). I would like to be able to use Truffle's HD Wallet Provider, and I need a mnemonic recovery phrase to do so. Is there a way to set this up for accounts on a private Ethereum chain?
When you generate an HD Wallet first you generate the mnemonic and then you generate the private key and address.
If you generated a private key by other means and you do not have a mnemonic, then it is almost impossible to find the mnemonic seed that will generate that private key.
You would have to brute force the algorithm PBKDF2, used by bip39 to derive the seed from the mnemonic. In that case certainly it is much easier to generate a new HD Wallet.
Generating a seed phrase is completely independent of where you're going to use it.