I imagine there's not enough room to legally build up the infrastructure that makes indoor plumbing viable

True; there's also the fact that the entire island is a county, and I don't know what they're counting as a 'housing unit'. Is it a dormitory used by researchers? Is it a permanent residence? Big Island is the entire above-water portion of five volcanoes, three of which are active, and I'd estimate that there's some housing used to house volcanologists and naturalists for research purposes, but I don't think even then that that would make a huge dent in the number of buildings without indoor plumbing.

Also, establishing plumbing on top of a volcano without care would probably be a problem. You don't want lava burning your butt while you're having a poop.

EDIT: There's also Kauai and Niihau! Niihau is a privately-owned island where a bunch of native Hawaiians are essentially doing it like their ancestors did, and Kauai has frighteningly high levels of economic inequality (the average house costs $528000 but the median household income is $62531).

Juneau and Anchorage should skew that data enough that there are a couple of white census areas.

Looking at the map of those areas Anchorage and Juneau are the only areas that aren't colored dark blue.