A code practice oscillator is typically connected to a straight key, and sounds off whenever the key is closed.
A paddle is typically connected to a keyer, which converts paddle input into dots and dashes. K1EL has an inexpensive keyer kit that provides and audio output.
MFJ may have fully assembled keyers that meet your needs. But, many keyers do not, as the radio transmitter provides the sidetone.
It may be possible to connect a code practice oscillator to be keyed by the output of a keyer.