Do I need a USB wireless internet adapter?

Depends on the WiFi chipset that the specific laptop has. And as these are usually not exchangeable, it either works or it doesn't. Get an external USB WiFI adapter, and you can choose one with (one of) the chipset(s) that work, and you may get an external antenna connection in the bargain.
Not sure about the VM though - direct hardware access from inside a VM can be somewhat tricky.

I honestly don't know why you couldn't pass-through the laptop's Wi-Fi adapter to the Kali VM. I've seen that too. However, even if you could do that, do you know if your laptop's Wi-Fi adapter supports packet injection? Some do, and some don't.

You're probably better off buying a compatible Wi-Fi adapter that supports packet injection. From there, you can pass it through to the VM and it should detect it just fine.