Simplest way to control a robot arm is with some microcontroller (board). For the web-camera, is this some wireless camera that will sent images to your PC? In that case, no problem, the microcontroller can take care of the control of all mechanical and electrical functions of the robot (charge control, robot arm positioning, ...). It can also communicate with your PC through a RS232 or USB-interface.
If you are new to microcontrollers, and wants to get started quick, you can buy some microcontroller board which comes with tutorials and has most functionality on board that you need for your robot arm. One board very well suited for you goal would be e.g. a **broken link removed** as it has motordrivers, servo-connectors, USB/serial communication and analog inputs on board. For the software-side, some nice tutorials can be found on the site.