You will need to do two things:
1) You will need a microcontroller that is powerful enough to run a web server of some kind. Typically this is a Raspberry Pi, or an Odroid, or a BeagleBone Black. On this microcontroller, run some web server software that can also communicate with the robot systems. Starting with node.js or Python SimpleHTTPServer might make things simpler.
2) You will need a way to address the robot. This means you need a static IP address that maps to the robot. If the robot is behind a firewall/router of some sort (which most things are these days) this means you need to set up port forwarding on that router/firewall to go to the robot. That, in turn, may need other network administration (allocating a static DHCP lease, etc.)