You certainly should be able to.
Since the existing motors are servo motors, not stepper motors, you will need a card for the computer which may be fairly expensive (about what you likely paid for the machine and the tooling).
And you may have to make some intermediate circuitry to feed the pulses from the Heidenhein scales into the computer. Other and later brands tend to output shaped square waves (two -- at a 90 degree phase shift) so you can count how far it moved, and in which direction. The older Heidenhein ones (such as some that I have) output a sine wave, and need special shaping circuitry before feeding it to the counting circuits.
O.K. You will have motors to move the axes. On many inexpensive home conversion projects, it will typically be using stepper motors, and the position will simply be inferred from counting the pulses fed to the steppers -- assuming that they never miss a step. This also applies to the early Bridgeport machines like the BOSS-3 Series I which has big heavy steppers, and mag amps to control the voltage to the steppers so higher speeds get higher voltages, and slower speeds get lower voltages to avoid overheating the windings. The higher speeds need the higher voltages to overcome the inductive characteristics of the windings.
Servo motors are a very low inertia DC motor, or special AC servos. The ones which I have experience with are the DC ones. Both have a tach generator so the speed can be monitored and controlled.
To move at a specified speed with steppers needs a train of pulses at the right intervals as long as it is moving.
To move at the same specified speed with servos, a command voltage from the computer via a D/A converter goes to the servo amp, which maintains the speed based on the tach feedback, and the computer can be busy doing other things, and only check every so often that the motion is correct. It does this by encoding scale which generate pulse trains with motion -- or with an encoder disc on the end of the motor. This is typically fed to a counter which can be read by the computer to verify that it is where it should be.
It is either 2-1/2 axis or 3-axis -- depending purely on the controller computer.
2-1/2 axis allows moving to a given Z axis position and then cutting normally in the other two, while 3 axis allows all three axes to be changing at the same time, resulting in very complex workpiece shapes.You can also add other axes such as rotary tables for even more complex operations.
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Good Luck, DoN.