Would a KVM switch do what you need? Or virtualizing the second machine's usb port in some way? Is there any kind of existing IPMI/ILO/etc functionality in the target machine that you could use?
If not: You need something that can act as a usb "device" (the "host" and "device" sides of a usb connection are very different). A search for "usb keyboard emulator" turns up a lot of projects in that area. I'm not sure you're going to get much simpler than a pi-zero or teensy, unless you can find someone selling a pre built device that meets your needs.
If you're lucky, your laptop has a usb port that can be configured to take the "device" role instead of the usual "host" role. In that case you could probably build a software-only solution using the Linux usb "gadget" framework to make that port act like a keyboard when attached to the target's usb port.