That's correct, when ARC is used the Windows tablet, laptop, or PC becomes the controlling device (wired or wireless).

The program is held on the computer side, the Open CM doesn't store the program.

is there a solution? because I need it to be used in competitions, the rules of the competition cannot have any connection so the robot has to walk alone

A solution may be to use mount a SBC (Single board computer) that runs Windows directly on the Robot itself. This is called an Embedded PC methodology.

There are a few examples of Windows SBC boards here.

If you need it to be a very small form factor I could recommend the Intel compute stick. They seem to be discontinued but many are available still on Amazon.

Note: When running an SBC directly on a robot, you will need a larger battery that can deliver a large number of amp-hours.

What’s the competition? A lot are very specific about the hardware and software they use.

All the robots in your competition will have an SBC installed to be the brains because microcontrollers like the OpenCM do not have enough power to be standalone - you'll have to do that same, as mentioned. A good and detailed tutorial in the getting started guide helps you pick a solution. Here's the link: https://synthiam.com/Support/Get-Started/how-to-make-a-robot/computer-configuration

As with @nink, I'm also curious about the competition. It sounds like that'll be a lot of fun. Of course, we'd love to see you win! Let us know how people here can help with your robot programming.