Of course they have goverment backing. They are doing work that belongs to the government: protecting citizens from foreign invaders. Furthermore, nothing would prevent politicians themselves or their family members from becoming victims, devoured by variants. So, the politicians would have not only official but also personal reasons to see the organisation is funded and supplied sufficiently.
That being said, the Japanese organisation's performance doesn't seem that good. Yui, a person belonging to an orgnisation all the way in Europe, and Kazu, who's officially nothing but a civilian, must already have worked hard to keep the town safe, defeating a number of variants. Kazu even had to clean the mess the Japanese organisation left behind in Susanoo's case. What did the Japanese orgnisation do in the meantime? They were only drinking tea and eating mochi in their billion dollar hideout paid by the taxpayers, I imagine.