I need to find some way to allow part of a project to swivel 360 degrees, continuously, while allowing 4 electrical connections to the base to remain uninterrupted. Is there some sort of swivel made for electronics that I could use?
If it is just four wires and carries little current, you might check the gizmos that keep telephone cords from twisting up.
Something as simple as a four-way 3.5mm jack and socket may do the trick.
Depends on how fast you want to rotate, and how susceptible to noise your payload is.
Well I have a story about these.
I was looking for a mains one (for an electric lazy sunday for mounting projects on and spinning with power to them) and I found a company called mercotac. I contacted them asking if I could have a sample, they said to contact local distributor. I contacted the distributor and they said that they could provide a single sample but I would have to pay unit price. Ok, I thought. It's only a connector it can't be too much.
£130!!!! >:(
Yeah. Not getting one of them then 
Mowcius
you might check the gizmos that keep telephone cords from twisting up.
Good idea, but shop carefully: most phone cables only use 2 conductors, so the cheap gizmos probably only connect the center pair. It might be necessary to buy a more expensive version to get 4 connections.
AWOL mentioned speed of rotation as something to consider, but the absolute number of times the subassembly will rotate matters, too: even if it's only turning at 1 or 2 RPM, if it's running 24/7, the rubbing will eventually wear out the contacts. It may be worth looking for lubricants that will conduct electricity.
A google search for "propeller clock" might turn up some ideas that other hobbyists have used to make cheap slip-ring connections.
Ran
mowcius
Automotive alternators usually inlude a slip ring for the field winding.
While I'm not suggesting you mount the whole unit, you certainly could use the necessary parts.
They have the following features :-
cheap
long life
high revs
high current
easily replacable brushes
In case you haven't put one back together, the brush holder often has a small hole that allows a piece of wire to be inserted to hold the brushes, while you put it together. Then pull the wire. 
Mark
mowcius
Automotive alternators usually inlude a slip ring for the field winding.
While I'm not suggesting you mount the whole unit, you certainly could use the necessary parts.
Oh I know of some solutions to the issue now but I just thoughr I would share that with you.
Mowcius
Some POV devices use this.
One used slip rings for power but could send data to it with and IR signal. I can't find a link for that one.
If your using slip rings, make sure your system is noise resistant. They have a habit of introducing a fair bit. (At least once they wear in).
The other option worth considering is a short IR link. A simple IR Led and a photodiode would carry the data across the gap for a lot less than a slip ring arrangement.
What is it for, if you dont mind me asking?
Cynar