Is this even possible? I need to connect multiple line level inputs into my machine, but my soundcard only has one Line-In. I'd like to use the Line-In on my onboard sound device to get an additional input
It is certainly possible to run more than one sound card in a PC - I've seen it done (but not done it myself) to achieve what you are looking for (multiple distinct input signals).
You may need two cards though, as your motherboard may disable its on-board card when it detects a dedicated one being present. Also, some cards drivers may not acknowledge more than one card at a time so you may need to do a little research to make sure what you plan is supported (this may be less of a problem with two different cards, but be careful as many cards have the same chipset which makes them effectively the same cards in some considerations).
OS and application support will be important - you need to add to your question what OS you are using and what apps you intend to use too if known.
You are likely to also find cards that support multiple independent input signals into a single card, though these will be aimed at a professional sound/music audience so are likely to cost a pretty penny.
What you need is an Mbox my friend. Windows will only use one sound card for line in.