[whym]

住民基本台帳収録変体仮名 (jūmin kihon daichō shūroku hentaigana, or Jūki-gana for short) is the character set specifically made for the administrative purposes of those local government registrars. I don't know what the actual computer system used there is based on, but it looks like Tron Code and Super Kanji support the character set. https://commons.wikimedia.org/wiki/Juki-Gana

The thing about hentaigana is that it's more or less an open set. I don't think there is the standard set of hentaigana that scholars agree upon. At a quick glance, juki-gana does not seem to be identical to Unicode's hentaigana set (https://www.unicode.org/charts/PDF/U1B000.pdf), although I'm sure the intersection is non-empty.