The specific law that Mitsuha mentions is called the 酒税法【しゅぜいほう】(shuzeihou) in Japanese. The Ministry of Justice does not presently have a translated version of this law (possibly because they haven't made translations as far back as 1953), but if it did, it would be located here.
In an article (in Japanese) about this specific scene, a lawyer (NISHIGUCHI Ryuuji) is asked about the legality of selling kuchikamizake. I have translated a part of the article:
According to Mr. Nishiguchi:
"Beverages with an alcohol content of more than 1% ABV are deemed 'alcoholic'.
"To produce alcoholic beverages, one must obtain a license from the director of the tax office within whose jurisdiction the production site falls. Producing alcoholic beverages without a license is a violation of the Liquor Tax Act.
"Article 54, Paragraph 1 of the Liquor Tax Act states: 'any person who produces an alcoholic beverage, yeast starter, or fermenting mash is subject to imprisonment with required labor for not more than 10 years, or a fine of not more than 1,000,000 yen'.
"If one were to produce kuchikamizake with an ethanol content of greater than 1%, it is possible that one would be subject to criminal penalties."
It appears that you can produce anything with an alcohol content of 1% or less without running afoul of the Liquor Tax Law. But some people produce umeshu [plum wine] in their homes. Wouldn't they be deemed to have produced an alcoholic beverage?
"There are exceptions.
"If you make homemade umeshu by adding ume [Japanese plum] to shochu, you are indeed deemed to have 'commingled' the alcoholic beverage with other substances and have 'produced' a new alcoholic beverage.
"However, consumers who prepare these beverages for their own personal consumption are deemed not to have 'produced' the beverage for the purposes of the Liquor Tax Law (Article 7; Article 43, Paragraph 11).
"That said, the production of kuchikamizake by fermenting rice would not be protected under this exception."