I would really encourage her to talk to her doctor about the vertigo. It may be quite treatable. But if it continues or gets worse she has to worry about
falls, causing an at-fault crash, or some other serious injury.
And I hear your frustration with your mother-in-law and husband for dismissing your concern. Vertigo (feeling like the world is spinning around you) may be something that has to be reported to the state licensing authority. Check the caring.com state-by-state guide to see what the requirements are for her state.
Instead of trying to persuade her, take a step back. You may want to try to discuss how she views her fitness to drive. I understand she relies on her car for her independence and income. Most of us do. But we are all apt to face a day when we have to transition from the driver's seat to the passenger's seat. No one looks forward to it. Has she any idea how she will know if she is no longer safe to drive? Raising family awareness about driving fitness may be a solid first step.