Vertigo/diziness is not a condition in itself, it is a symptom that may indicate various other conditions, not all of which could be diagnosed by a blood test.
Low blood pressure, low blood sugar, iron deficiency, an ear infection, menieres disease, cardiac issues, a tumor, etc, etc. can all cause vertigo/diziness.
Your GP is more likely to perform observational tests than a blood test, they may do this whilst simply talking to you and observing you.
A blood test would only be testing for blood sugar or an iron deficiency, which may be obvious from observation anway.
If your GP does not believe that your vertigo/diziness is serious enough to prevent you from working then he will not give you a fit note.
You can always go to another GP for a second opinion.
https://www.nhs.uk/conditions/vertigo/
https://www.nhs.uk/conditions/dizziness/