Turnkey supports Doge address derivation with ADDRESS_FORMAT_DOGE_MAINNET and ADDRESS_FORMAT_DOGE_TESTNET address formats. Doge addresses are derived from the secp256k1 curve, which Turnkey fully supports.
Turnkey supports Doge transaction signing through the core signing capabilities.
Check out the examples/with-doge respository that demonstrates how to construct, sign and broadcast a Doge P2PKH transaction on Testnet.