Is it possible to refuse everything is a big question, but, for example, dish detergent is an excellent substitute for dry mustard. Simple and environmentally friendly, cleans as well as detergent. On the net they write that it is also suitable for washing sinks and kitchen surfaces, but I personally have not tried it yet.
You need to go back to the 18-19th century, when life was much more "ecological" and short-lived. The linen was boiled in huge vats, and the laundresses could be distinguished by their burnt, crimson hands. Dirty, greasy dishes were rubbed with sand and washed with water from the river, where they poured everything that fell. No wonder so many people died from cholera. Washing in the ice hole in winter is an adventure! Cockroaches and other insects were killed by not burning the stove for a couple of days and freezing the whole house ... In general, I doubt that with such wild tourist costs, the achieved level of cleanliness will be sufficient for you. I also have a hard time imagining how modern equipment, such as a washing machine or a dishwasher, can work without accompanying household chemicals.
It depends on what is meant by household chemicals. If this includes laundry soap, soda, vinegar, ammonia, then it is unrealistic to completely refuse.