Would a radiology degree allow someone to be a sonographer?

There really isn't a such thing as a "degree" in radiology. You would be a radiologic tech which is the person who takes the images or you would be a radiologist which is a doctor.

There are some sonography programs which require you to be an accredited radiologic tech first, however that is not always the case. Do a quick google search and you can find some sonography programs.

If you want to be an actual radiologist you will need a bachelors degree, medical school, plus 5 years of residency.