groundfungus:
To do what you want without sensors, you will need stepper motors.
Using a stepper motor would remove the need for wheel encoders, but with wheel encoders as proposed you could use any type of motor.
You will need to manage the acceleration of the 'bot to keep wheel slip acceptably low. Just limiting the maximum acceleration and deceleration should be sufficient. If you decide to use stepper motors, you can take advantage of the accel stepper library to manage that for you.