Only if being able to drive was a required as part of OP’s duties. If it isn’t, it is not relevant to her fitness to work.
I am confused by posters who think that if OP is too drowsy to drive to work and back then she would be entitled to be off sick. This is despite the fact the OP has said she is capable of doing her job, just not ok to drive. If she is fit to do her job, she should not be off sick.
Not being able to drive to work does not mean she is not fit for work. If it were, then anyone who does not have a driver’s license, is temporarily unable to drive, or who does not have a car would be entitled to forego employment. Which is clearly not the case.
Her fitness to drive is irrelevant to her employer- they are probably only interested in whether she is fit to complete her duties at work, or not. How she gets there is not their concern or responsibility. If she cannot drive, she will need to look into other means of transport to and from work. Just like everyone else.
The only reason to be signed off as unfit to work is, well, being unfit to do your job. OP says she can do her job, she would prefer to wfh to avoid transport issues, which is categorically not the same as being unfit to work.
It’s fine to ask her employer for flexibility, and it’s fine to for her employer to want more information or proof before deciding what to do. It’s not fine to pretend to be too sick to work because not being able to drive is inconvenient when it comes to commuting.
If a patient asked me for a fit note to sign them off work on health grounds because they could not drive, where a requirement to drive/operate machinery/other safety critical responsibilities is not part of their duties, I would not be able to state they were too unwell to work. It would be a false assertion.
If the patient said that they were too drowsy to do the essential tasks for their job, I could sign them off sick for that reason. But those are two very different scenarios.