No currently a P46(car) can only be submitted for 2024-2025 now.
HMRC (the Government Gateway) only supports the 'a current year' - so currently that is 2023-2024 for P11D returns, and 2024-2025 for P46(car) returns. You can't send P46(car) submissions for a past tax year (e.g. 2023-2024) any more - the deadline for electronic submissions was the 5th of April 2024 - after this date the Government Gateway is 'rolled on'.
In theory you could still print out and post paper P46(car) returns for that year, but we would expect those to take longer to process at HMRC than simply waiting for the P11D submission given current circumstances.
The fastest way for an employee to affect their tax code now would be a personal representation, either through their personal tax account (web) or via a phone call.