[JSeymourATL]

A trick Tim Ferriss (4HWW) employed was to befriend very specific bloggers and targeted authors who frequently wrote about aligned subject matter. He would very causally offer to share tidbits of what he was working on, sensing it would pique their interests. Steadily he was able to build buzz and street-cred via a non-traditional route. Quite brilliant strategy actually.

[CyberFonic]

Your idea can be equally applied to academics. As suggested elsewhere in the replies, you could identify academics who are already working in the relevant field by doing a bit of background reading. Then use carefully use the Tim Ferriss trick.

Academic institutions are, generally, keen to be seen to be collaborating with industry. So if said joe can work that angle there might even be collaborative research grants available to the cooperating institution.

As mentioned in my reply, it might be prudent to establish priority by self-publishing on average-joe.com and a reputable internet scientific archive like ArXiv. Doing so mitigates the risk of somebody else claiming the discovery as their own.