From your description, it sounds like the device you want to connect presents itself as a virtual COM port when connected via an associated Windows or Linux driver. If that is the case then it is very likely the USB Host Shield will allow you to connect that device to the Arduino.
The important thing to understand is that the Virtual COM port is a feature of the USB Driver in the host. With the host shield, you have to make your own drivers. This means making a driver which talks to a specific USB serial device. The device may be of a number of types (CDC, Prolific, FTDI, ..) and you driver will have to be written for that specific device.
First check out the circuits@home web site and use the described there utilities under Windows or on the host shield to find details of descriptors fo your USB device. If the device is one of the common documented types, then a driver should be no problem. I put some code on circuits@home for driving the FDTI device already.
Let us know when you have the device descriptors.