When Adapting an EF-s Lens to my R6 it automatically switches to only capturing APS-C Format on it's sensor. Is there any possibility to override this and capture the full sensor? Even though it might not be all covered by the lens or have heavy vignetting?
According to the Advanced Users Guide for the R6, page 129, "With EF-S lenses, [1.6x (crop)] is set automatically, and no other option is available."
Not with Canon's own adapter. Perhaps with a third party adapter?
An alternate way is if you covered all the electrical contacts on the lens, so the camera doesn't know what is attached. You'd need to focus manually, and you're stuck at wide open. You also need to setup the camera to allow it to take pictures without a lens attached.
But as mentioned in another answer, the lens isn’t designed for this usage, so you will probably not get a good image. For a zoom lens you may be able to fill the frame on the narrow end, but most probably it will be very soft, vignetted, and distorted in the corners.
As an example, I have been using the Tokina 11-16mm f/2.8 DXII lens on my R5. It's a crop frame lens but with an EF mount, so the camera treats it as a full-frame lens. At 11mm I only see a circle, but at 16mm it fills the frame. It's sharp in the center, but quite a lot of COMA distortion at the edges, and particularly the corners. This night shot was actually taken with the lens at 16mm f/4 (stopped down to try and reduce the COMA with limited success)
The issue is that an EF-S lens, adapted or not, is going to project a smaller image circle than an adapted EF lens or a native RF lens - one that is suitable for the smaller APS-C sensor that camera bodies designed for the EF-S lens will have installed. Even if you could override the settings to use the full sensor in your R6, that image circle would still not cover the entire sensor, resulting in the appearance of severe vignetting, and probably other aberrations especially toward the limits of the image circle. Overriding that will not actually allow you to use the full sensor to capture a larger image.