I've already mentioned LiveImpact as a resource for building and maintaining a website. But their whole pitch is an all-in-one non-profit platform, so they do a lot more than that. If you are just looking for a web site, you can still check them out. However there are other more narrowly-focused options.
If you are small and just starting out, the best way to go is to sign up for one of the do-it-yourself (DIY) site builder services. These have gotten pretty good lately, and though they are ultimately limited in what they can do, they still can do a lot (mobile responsive, shopping cart/commerce...) SquareSpace, Yola, and WordPress.com are a few of the more popular ones. (I've used Wix and I do not recommend it.) Of course you or someone in your org will have to have some basic design and technical skills.
Also—while I haven't had first-hand experience—some of my colleagues have had good luck hiring people via online freelance marketplaces. There are some very talented people living in low cost-of-living areas that will be cheaper as long as you don’t mind working with them entirely online. UpWork is the biggest one. Guru and Toptal are alternatives.
Next you could go ahead and hire an agency. That will give you the most flexibility but it is a more expensive option. We use Visceral at my org. Here are some others that look promising (but I have no experience with them):
Finally, you could just hire someone and do it in-house. If you've gotten to this point in your org, you've probably outgrown this blog.