Can anyone recommend a reliable, affordable website design company that understands non-profits?

You might want to visit https://www.wildapricot.com. They offer sevices for membership organizations and they have a free plan that should enable you to access their website templates. This way, your volunteers could transfer your information to a a template you choose for your non-profit and I believe there would be no set up or monthly costs (unless you wanted to upgrade to services beyond their free plan in the future).

Good morning. We can definitely help you out with a new design and most likely integration with any systems you already use for donation, email marketing, etc. I've personally built several nonprofit websites with varying amounts of information and functionality.

I am surprised that you do not have a volunteer in who could help you.  Websites today are much easier to put together then every before and if you are unfamiliar with sites you would most likely not really know that.  Another option to consider is doing it yourself.  The design and the software of today makes it very possible.  It might take a little learning which might take a little time but since you have waited this long a little longer will not hurt much.

Yes! I specialize in nonprofits but use platforms whereby organizations are setup to do their own editing, saving hundreds if not thousands of dollars!. Message me!

Russ Haskell with www.alphaandomega.com has done many non-profits and church web sites, all done better than good. 

I'm on the board of an all-volunteer non-profit that uses Weebly.com. It's not free, but it's cheap and easy to use, and it's scalable. For example, we recently decided to upgrade our membership so that we could add e-commerce, and it went very smoothly.  If you want to see an example of what you can do with Weebly, our website is greenvillehiking.com. 

Absolutely!!!!  Rakacreative.com

Grassroots Music Foundation's national coordinator/ Barbara Milhem.