Hi Aidan:
If this is something that your mother in law has never experienced before and isn't currently under the care of a doctor for, I would strongly suggest that she go to the ER.
Dizziness and weakness in her left arm could be indicative of a heart attack or stroke. I'm glad that you care enough to ask about her. Again, I suggest the ER as many women have heart attacks without symptoms.
Has she felt nauseated, more tired than usual as well. If so, she should really be seen as soon as possible.
If I can help you further, please let me know. With your post, your question reads should she go to the ER or DR? With the information that is provided I have to say the ER.
If this has been helpful, please do accept. Positive feedback is also welcome.
Good Luck!
Terri L. Romaker