One thing to understand about nonprofits is that although most now recognize that they now need tech, they (rightly) do not generally consider it part of their core mission.
Thus instead of keeping someone on staff, they outsource technical work to agencies. (Just as they outsource other important but non-core functions, such as developing, printing, and sending all that snail mail spam they send every winter.) There are even a few agencies that specialize in tech work for nonprofits. Other organizations use services that offer a hosted CMS and fundraising database and other tools in a single package along with high-touch customer service. Small nonprofits may hire freelancers to manage their website for a few hours per month, and try to run their own email campaigns.
Larger organizations will often have someone in house whose job is to plan and manage their digital presence and campaigns. That person is generally a tech-savvy layperson. Their job is mostly to communicate between internal stakeholders and the external providers.
Other organizations may be data-heavy. They may have a DBA/analyst on staff to handle data management and reporting.
A very small subset of nonprofits see technology as core to their mission. These are probably the ones you're looking for. Just keep in mind that many of these are relatively new kids on the block, so to speak. Few of them have the resources to hire more than one or two technologists, thus those people are of necessity very broad generalists.
A subset of well-funded major institutions have decided that data or technology is part of their core mission or need internal tooling beyond classic IT desktop and network provisioning. Institutions of higher education come to mind. So does the NYPL, which has all sorts of fun digital initiatives.
Another possibility you might consider would be government sector work. There are hundreds of federal agencies that need everything from websites to data management to payroll systems, and a few doing fun things with open data. There are also state and local agencies - from transit to child welfare agencies - that need similar things on a smaller scale; some of which are doing very good things, and whose efficiency you might help improve.