The best way is to buy a quality flat bed scanner (actually the *best* would be a microfilm camera). But if you're a volunteer and you'd have to pay for the equipment to do your job, I'd suggest going for second best. Do you have a decent smartphone? They take pretty good pictures. There are even some apps that make the camera work more like a scanner, though I'm not sure exactly what they do (Ancestry has a free "shoebox" app but I haven't gotten it to work). What some can do is to merge multiple pictures together into one doc, if it's too big for one clear picture. It takes some practice to get photos to look as good as a scan, but it's doable. I suggest investing in a high quality sheet of plexiglass to put over the newspapers to keep them flat as you photograph them. Experiment with lighting a lot to reduce any glare (standard overhead lights may not work).
Once you have the paper digitized, you'll need software to crop each picture appropriately and then to paste it all together. Photoshop is the gold standard here and maybe the library already has it or would be willing to get it. But there are other cheaper ones, even the occasional freebie. A resource to check out is Techsoup.com, which is an awesome source for nonprofits to get software, including Adobe/Photoshop, at very low prices. Photoshop is likely overkill for you; all you really need is cropping (which any software will do at equivalent quality) and pasting (ditto) but it would be helpful to have some tools for changing the lighting, sharpening focus, etc.
Instead of printing it all out, I'd get PDF creation software (Abode Acrobat works but there are cheaper ones) and turn each newspaper into a PDF doc with multiple pages (or do each item separately). Put it on a computer accessed only at the library or put it online for anyone (with or without a library card) to use.
Note that there are places that have inexpensive hardware as well (TechSoup has a bit) and you might be able to get a scanner for cheap or even free. Would you be allowed to hold a small fundraising event for the hardware/software you need?
Good luck, this is a fabulous project and I can tell you it will be of great interest to people beyond your local area.
Cyndi