Do you want to add an Arduino to your robot? Are you going to interface your Arduino with the PIC? How (I2C, Tx/Rx)? If this is the case, you have way too many challenges in your hands. I would completely drop the PIC and replace it with the Arduino. Or get an Arduino-controlled robot. Have you ever used an Arduino before?
Before you go into details (controller chip etc), you need to understand the big picture. As I said, Arduino+WiFi shield could act as a web server that processes requests from your laptop (connected to the network) or smart phone (connected to the network through wifi). You could test this setup without your vacuum cleaner in place, just to check how the commands get through. Then add your relay board to the arduino and modify the sketch to include them. What kind of vacuum cleaner are you using? A roomba?
CrossRoads, Cool board, I like it. An actual LED project using it would give some inspiration to potential customers. The TPIC595 seem to be able to drive multiple LEDs in series/parallel as long as the current does not exceed 100mA and the voltage is no more than 33V. Keep up the great work!
One way the vacuum cleaner can be controlled by your Arduino is through a relay. You can basically turn it on and off. Moving the vacuum cleaner around is a bit more challenging. To send the commands to Arduino through the internet, you would need to run a web server on your it, either using the WiFi shield or the Ethernet shield. WiFi shield would make the device portable, if this is what you need. As Arrch said, you should give more details on your project.
From your code, I see you are trying to make an http request (GET). Maybe the port number is not (the default) 80, so it needs to be specified. The first step in debugging your code is to get a connection to the web server through your browser. Is that your IP address? Are you running a web server on it?