The advance of the cashless society, which is slower in Japan than many places but finally happening, poses a particular practical problem for jinja.
When one visits a jinja to pay one’s respects to the kami, one is supposed to make an offering. On an ordinary visit, it would just be a few yen (a few cents), often five yen, throwing some small change into the offertory box. For a formal prayer, the offering must normally be at least ¥5,000, with some jinja setting ¥10,000 as a minimum, but it is an offering to the kami, not a payment to the priests for the prayer. That is why the money is supposed to be put in a formal gift envelope, with “Hatsuhoryō” (First Fruits Money) or “Tamagushiryō” (Tamagushi money) written on the front, and handed to the priests before the ceremony starts so that it can be offered to the kami as part of the ceremony.
Jinja already have problems with people just taking money out of their wallets and giving it to the priests. It is not so much that this is a breach of etiquette, as that it represents a fundamental misunderstanding of what the money is. It only makes sense to do that if you think of the money as a payment to the priests for their services; an offering to the kami has to be in a form that can be offered, and a handful of banknotes does not really meet that. A number of jinja supply simple envelopes, so that the notes can at least be put away.
Obviously, cashless payment makes this even more difficult. Most jinja cannot take cashless payments at the moment, but getting that set up is by far the lesser problem. The important issue is that, with a cashless payment, there is nothing to offer to the kami.
The nature of the offerings has changed over time. At the beginnings of Shinto, people offered food and cloth, or crafted items. As the cash economy advanced, cash became the normal offering, but it was always regarded as a replacement for the food or cloth that would be offered.
This suggests a way for jinja to resolve the issue. They could have small, symbolic amounts of rice or cloth that people could “buy” for ¥5,000, and then offer to the kami. The purchase would be done using cashless payment, and the purchased item offered during the ceremony. This would actually make it practical to return to some older traditions, such as offering cloth supported by a wooden wand.
If jinja take this approach, they do have to prepare the offerings, and the offerings need to be really cheap without looking really cheap. (Using the same item as the offering for multiple ceremonies is symbolically a bad idea.) I do not think that either of these problems is insoluble, but jinja are going to need to do something along these lines in the future, or simply accept the money as payment to the priests.