Is it normal that a PhD supervisor requires a student to apply for a grant?
So, first of all, in Europe the relation PhD sutudent–supervisor can be detached from the relation employee–employer.

While it is possible to obtain a PhD sitting on a different job all the time, I can only support the urge to move to a grant-funded position. You would have much more time for your research when compared to part-timing for money. I am speaking out of experience.

So, the first conclusion is: you probably need a grant.

Now, for application. Typical funding bodies I know, require the applicant to hold a PhD already. So, it's your supervisor, who needs to apply. Contrarily, a stipend can be granted for a student. I heavily suggest talking in person about this issue.

The second conclusion is: you need to clarify, what kind of funding it is and who would do most of the writing. Talk with your supervisor.

The general rule of thumb is: agree with your supervisor, as he/she has much more experience in the academic questions than you do. There are some common pitfalls and non-obvious issues. For example: where to publish, what kind of a position is better, etc.