You have the same exact problem that all astronomical telescope drives have. The easiest solution is to align your axis so that one of them is EXACTLY parallel to the Earth's pole. Now you only need one motor to track the sun as it moves.
The second motor needs only a small range of motion and only needs to move a tiny bit every day and then it can "put on the brakes" literally to keep this axis from moving. We call this the declination axis
This type of drive moves in something like a latitude and longitude coordinate system. They are spherical not cartesian and it make tracking the sun very simple. So simple you don't need computer control.
The large motor needs to rotate about one revolution per day. The smaller motor makes a few degrees of motion per year and then reverses. It is mostly stationary.
You are in fact building a CLOCK and in fact in these these drives were called "clock drives" because SPEED is the thing you need to control not position. The sun goes not move randomly. A hundred years ago, before electronics were available, they used springs and pendulums to regulate the speed, just like you'd see in a large clock.
typically the axis that is parallel with Earth's pole called the "polar axis" is tilted to match your latitude is fitted with a large worm gear. Get the largest and best one you can afford. The pinion that drives this worm uses a motor that can be very accurately speed controlled. It will need an encoder on the motor shaft to measure speed. You will need a feedback loop to keep the speed constant as the mechanical load changes.
In any case think more about controlling exact SPEED as this drive will move very slowly and must not do so in steps or you will have huge vibration problems. If the mass of the load is high you MUST move it continuously not in anything like small steps.
One other thing: You seem to want to move this "open loop". But why not try and track the sun with some kind of sensor? All you need is a small tube with both ends closed. Face one end approximately at the sun and make a tiny pinhole in the sun facing cover. this will project a tiny dot of light on the other cover. control the speed of your polar axis drive motor so as to keep the projected dot from moving. Someone suggested placing an encoder on the shaft, no need when your target is so bright and easy to track.
Problems you will face: (1) your axis is not perfectly aligned with the Earth's axis. This means that for sure you can not use an open lop drive and WILL need to track the Sun optically. This is another reason not to depend on a shaft mounted encoder. Also I lied about the declination motor being able to clamp on the brakes. That only works in the axis are perfect. It will not be perfect.
You will have these problems but even worse it you try to build an "X, Y" type drive where one axis is vertical and the other horizontal. You will find that your two axis are not at exactly 90 degrees and they are not pointing where you think they are. You will have to track the sun with a sun sensor less you spend a LOT of money building a very rigid and precise mechanical system
the good news is that this is a well known problem and there are many ways to go. Computers make it cheaper then in the last century because we can build a cheap sun tracker to compensate for the faults of a less then perfect mechanical system.