Can I use USB data pins as a power source?
This is neither Kosher nor Halal. It should actually be explicitly forbidden in the Electrical Engineering code of ethics. Some certification standards explicitly frown upon this.
Besides the wires probably being extremely under-dimensioned for the delivery of power, using narrowly-defined standard connectors for non-standard uses is a sure way to cause an unforeseen problem down the road. Particularly if this use would ensure the destruction of any standards-compliant device that could be plugged into it, as is this case.
I have had apparently standard but mis-wired cables in my hand that have caused the malfunction and even destruction of standard-compliant devices.
Standardization bodies actually spend a lot of time designing their connectors so that these cannot be confused/mistaken/interconnected with anything else out there. Don't make our job harder.
Some connector types are more generic than others, stick to those.
If this is not possible, is there an existing type of cable that can achieve this (3 power leads and a ground), or will I have to solder my own?
Cables will do whatever you want them to do, they carry current from one point to another. The biggest concern would be if someone accidentally plugged the USB cable into a regular USB device which would probably blow out the data lines on that device.
Most USB cables use bigger conductors for the power lines, so the data lines would best be used for lower currents.
But you can use any cable you want as long as you can find ends for it and a way to solder your circuit to the ends. At that point it might be easier to just go down to your local hardware store and splice a 4 conductor cable and solder to that.
Your best bet is likely to be a more generic connector. There are many common types of cable out there other than USB that will have two major advantages: Larger conductors on all pins, and no chance of damaging USB devices by mistaken connections.
Consider using a DIN connector on either end of your cable, for example. These connectors are common enough that they're cheap, and generic enough that the presence of the connector does not automatically make people expect it will work for any one protocol.
Molex's mini-fit and micro-fit connectors are also popular for power, but be careful if you use mini-fit jr connectors, as they are commonly used on PC power supplies.
JST connectors are commonly used for batteries in RC applications, and may be a good fit for this as four-conductor JST connectors are sufficiently uncommon that people wouldn't have any particular expectation for what they're meant to be used in, yet sufficiently common that they're very cheap, even pre-assembled cables with a four-pin JST connector on either end can go for less than $2.
The links in this post are just what I found in a very quick search of digi-key; if you search yourself you may find more options, and you can tailor to your needs regarding conductor size. Or you can look for just the connectors and make your own cable, which is not that hard if you have a crimping tool.