How to solicit funded research projects from industry to perform while in graduate school?
How would one go about this assuming no contacts are known a priori.


Likely you can't go for this without any contacts at all. You are thinking about this in the wrong order - you can't get money from industry first and use this to build up contacts to industry; rather, you would build up contacts to industry to achieve a level of trust so that you would be later on in a position to receive industry funding. This also means that getting direct industry sponsorship as a PhD student is at least uncommon - typically, companies are more interested in sponsoring a professor or chair that they have positively interacted with over a sustained period of time.

Slightly longish story short, your planned approach (acquiring industry funding) is not a suitable means to your stated goal (landing a job in industry).