Yes, you'll need to connect the video output of your laptop and a USB cable for the mouse and a USB cable for the keyboard to the KVM switch.
So yes, that is a lot of cables. But you can buy a single cable that has all the necessary plugs on both ends. While it reduces the amount of cables, those cables are usually ugly to see, a lot stiffer than normal cables and only available at certain lengths, so for one system the cable is too long, for another perhaps too short. Those cables can be bought via Amazon if all those problems are not an issue for you.
The webcam will be an issue. At least I have not seen a KVM switch that supports a webcam. Keyboard Video Mouse is where KVM stands for.
KVM switches usually do not need drivers, Some may come with software to manage these if you buy a fancy model. A standard 4 computer KVM switch is already quite expensive. Fancy models are very expensive.
In 2 different companies I have worked with KVM switches. In the first job, I got to use a fancy one for 4 computers (but only 3 connected). That was a nice one and at the time more expensive than a decent desktop. In the other job I got a very old-fashioned and basic one. Very unreliable, as is was using the PS2 ports for mouse and keyboard. With both KVM switches no drivers were needed.
Managing the video, keyboard and mouse from 4 laptops via one USB-C cable is in principle not a problem. But you'll need USB-C cables that actually can transport all those signals. And no, USB-C cables are made according to very different standards and even if the packaging states it support transfer of signals you need, you better check as those manufacturers are more often than not less than truthful. Expect to pay dearly for USB-C cables that are actually capable.
Oh yeah, you'll need laptops with USB-C ports that can handle all those signals as well, so you end up buying 1 or 2 or perhaps 4 laptops to be able to use a single USB-C cable KVM switch solution. And then we'll probably also need a new KVM switch that works with USB-C as well. Blame the governing body behind the USB standards and the way they give leeway towards device manufacturers with their vague/unfinished requirements for the mess that is USB in general and USB-C in particular.
For a single cable solution, there is also the Ethernet KVM switch. Not cheap either and you'll need one for each laptop, but it works through a sigle Ethernet cable. These do require drivers and are not as easy to work with as traditional KVM switches.
If all of the above sounds complicated, expensive or both, you can also use a piece of freeware, called: Barrier
You'll need to setup one laptop as the master and the others as slaves. And all laptops need to be in the same network segment. That is not a problem in your home network, it could be in AD environments. The way you have to configure which laptop is which, may be a bit weird the first time you'll need to do it. But once you get your head around it, it is actually easy.
With the Barrier software you'll still need to screens from the 4 laptops visible to you. When everything is configured, you can drag the mouse cursor to any of the 4 laptops and use the mouse/keyboard/trackpad from the master laptop to type on the laptop the mouse cursor currently resides. If all laptops use Windows, it should also not be a problem to have a shared clipboard between all laptops. I personally use Barrier between a Windows 11 laptop and a Linux laptop and the shared clipboard feature barely works when the Linux laptop is the master. It doesn't work at all when the Windows 11 laptop is the master. But that is only an issue in a mixed environment like mine.
Barrier is not a full KVM switch solution, because it doesn't do video, only mouse and keyboard. But it is free, very usable (once you have seen it in action) and besides a bit of time getting your head around the configuration screen, a solution that can help you out almost immediately. Best of all, it doesn't require any extra cables. Lets just say that since I know about software like Barrier, I rather spend money on a larger desk and possibly an extra monitor for a desktop, than on any KVM switch solution.