A lot of the related questions require variables that we simply do not have. I would like to calculate sensor size for a video camera. We have focal length and resolution but I do not know the field of view or crop factor or any other additional camera information. Is there a way to calculate just from this? If I have to have something like field of view angle, can I calculate this without sensor size and then use it to find sensor size?
You cannot calculate sensor size from just focal length and resolution. Since resolution can vary depending on the sensor's construction, this just plain is not enough information.
If you have a scientific application, you should buy parts where the specifications are known. If it doesn't matter that much, you can measure the angle of view in practice from sample images of known objects, and then, since you know the focal length, work back from there.