Your measurements are in a log scale, and Common Loudness is a log scale, so the conversion is linear. Just pick a Common Loudness value that you think corresponds to dead-of-night and one that you think corresponds to a loud action scene. Then, it's simple algebra to interpolate between them. Whatever bias may exist in your sensor gets removed, because your results will be relative to the endpoints. Actually, you only need one point to remove bias; having two endpoints also removes scaling error.
As Hobbes and Mark point out, non-linearities in the sensor can significantly distort your results. As Hobbes mentions, clipping can be a big factor. If the measurements seem to flatten out when the sound gets very loud, then you need to take the high calibration point at the knee of that curve and consider measurements above that point useless.
If the camera is giving you the measurements directly, then I would expect the measurements to be taken before AGC is applied. If you are feeding the camera audio into some other device to take the measurement, then AGC will completely distort the results.
Note that the transfer function of your webcam might not be flat, meaning that sounds at different pitches might not be measured on the same scale. A room full of screaming children and a charging herd of cattle might sound like the same level to you and me but be measured at very different levels by your camera.