the only reason you need it hooked to your computer is for a place to store the information.
if you find another place to store the video, its as simple as that.
it all depends on what you want to do why you want to do it, and how secure you want it.
if you want your camera (or web-cam) hooked up directly to the internet (a seperate server) you could buy a wireless camera that hooks up to a home site. these however are extremely easy for others to view, (you can view them by searching the IP in google :/ ) the plus side, is you can view it from anywhere aswell ( work, school etc. online)
you could go oldschool and get a camera hooked up to a VCR, then when you come back and view the tape, simply fast-forward through it to see if it caught anything, then over-write it if their isnt anything. downside is if whoever is being recorded by this can usually follow the wires to the VCR and just take the tape :/
if you're not looking to replace your camera (which it seems you dont)
what kind of webcam is it?, if it has to be hooked up to your computer, then not much luck there you'd have to replace it (unless theirs a wireless usb transmitter i dont know about lol)
do you want the feed to be loaded on your computer or else ware for future viewing?
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sorry i didnt see your message saying you wanted it on an external device
what type of device?
a simple usb hard-drive?
if thats the case i do believe that the hard-drive doesn't deliver any power, so the camera thus wouldnt work
i'd need further clarification on what you have and where you plan on storing the video in-order to help :/