The last time I did this was something like 25 years ago.
From what I remember I did not have to take the neck off. The pickguard should fit around the base of the neck, but it may not be easy to fit it in unless the strings are removed. It should be obvious that you take off the old guard, but remember that all 3 pickups and all 3 knobs have to removed from the old pickguard and re-installed on the new pickguard before you put it in place and screw it down. I did remember that the pickups and knobs, once removed from the old guard, are somewhat of a tangled mess that you will have to manage. Also marked which pickup came from which position in the pickguard since they are different.
One last thing about aesthetics. There are at least 2 different Strat pickguard configurations that I know of, the difference is where the pickguard screw closest to the middle pickup is located. Most of them are right next to the middle pickups, but some are located higher up on the pickguard. If you like to get the details right make sure the pickguard you buy has the same screw position pattern as your existing one.