Putting aside the grant funding idea, I think you could really democratize academic publishing by adding a StackExchange-like rating system to something like ArXiv, and also building in a conference organization system (this part is actually key). You'd allow researchers to link off-site (even paywalled) content to their profiles (email to the address listed in the author list should be sufficient authentication), and bootstrap the rating system by granting reputation points in certain subfields to authors with recent publications in reputable conferences/journals. You get reputation by people who already have it positively reviewing your papers. The system gets papers by including a really good conference organization system (in-person networking is still very worthwhile, even if we solve the journal problems of distribution and peer review), which requires conference submissions to be put into the system, along with the reviewers comments. You'd fund the thing (and weed out junk submissions) by imposing a relatively nominal fee (say $50) for lifetime hosting.
Such a system is technically possible, but it would need to achieve critical mass to actually replace the existing journal system.
(I should mention I'm coming at this from the perspective of a computer science researcher - in a lot of the subfields of CS the most expensive parts of doing research are 1) your time and 2) institutional library access to get at the papers you need (Google Scholar helps here, but it's far from perfect), so large grants are less essential than in disciplines that need expensive laboratory equipment or travel to exotic locations.)