If you are using iCam/icamsource, a computer has to be running icamsource 24/7 if you want to be able to access the feed at any time.
If you are asking in just general terms, it depends on the type of webcam you have. USB webcams are hooked to a computer and get their power from the computer...so obviously the computer must be turned on. If you have an IP camera (generally not called webcams), then it has a power adapter and can be accessed anytime as long as your internet connection is ok and your router is turned on. And no matter what, you obviously need an internet/data connection on your travel computer (or iphone/ipod) in order to connect to your home IP camera or computer (hosting the USB camera).
If you don't have an iPhone or iPad with a 3g connection (only a computer), then you'd probably be better off using icamsource to detect and capture motion events...and then use dropbox (free) to backup the motion events on an offsite server. Then you could access these files using dropbox.com's website. It's a bit of a hassle to set up port forwarding, picking the right program to stream your webcam feed, etc...so this would be the easiest thing to do by far. Of course, SKJM made icamsource as a support program for the iCam app (which is how they make their money)...so it'd be nice if you threw $5 their way by purchasing the iCam app (even if you can't use it)...or maybe buy it for a friend. Search on here for dropbox and you'll find some directions on how to use icamsource and dropbox together. Good stuff!