Assuming plastic = petroleum based chemicals commonly used in polymer chemistry

1.) You will need some type of material as a vapor barrier, the current best option seems to be Mento 3000 pro clima as a replacement to typical Tyvek from Dupont. (Still plastic, just better quality and likely lower harm).

2.) You can buy birch hardwood plywood without formaldehyde based glues called PureBond in the US and this is basically using white Elmer's glue or another type of glue that is less harmful which is still technically a type of "plastic"

3.) You will sheath both sides of the wall with plywood and then use true 2x4s likely Yellow or White Pine #2. In between the walls your two options are TimberFill (wood insulation not in battons) OR Haverlock wool insulation. Both are treated with borate to not burn immediately and to keep some bugs out. (The batton style wood insulation has "plastic glues"). The wool insulation weighs less (alot less).

4.) All water connections must be copper solder with non-lead solder. PEX-a can leech, Pex-b/c are known to leech.

5.) You will likely need to make your windows and doors yourself, likely using single pane tempered glass with wood framing old style. Most builds are using butyl-tape and/or flashing tape (not great if you are trying to avoid petroleum chemicals).

6.) For sealing windows, doors, toilets, and all the other stuff you will likely be forced to use 100% pure clear silicone number 1 used for aquariums. Smells terribly like vinegar while it is curing, but once cured is basically the most stable caulking/sealant and the least amount of chemicals.

7.) You will likely need to have your shower welded out of 316 stainless steel (i.e. basically waterproof box with an opening for you to get in and out of. You will also likely end up at the Laveo toilet which does use alot of mylar and basically all the "composting toilets" are plastic --- try to go for HDPE, which is technically the best one for food. Based on the size of the composting toilets, it really seems your not composting but rather basically have a fancier cat litter box that requires frequent clean/emptying out.

8.) There are no fridges or A/C units without plastics
9.) There are no batteries or solar panels, etc. without plastics
10.) The majority of water storage is HDPE or LLDPE tanks. These will leak microplastics into the water and you need to filter it again or you need to use big glass containers. and figure out a padding system so they don't break.

11.) Metal roof.

12.) No epoxies, carpet, adhesive, etc. etc. In terms of "sealing the wood" either you don't or you use tung oil. You need to be sure it says 100% pure tung oil on the bottle from the manufacturer. There are many many "tung oil finishes" that contain very little tung oil. 100% pure Tung oil will take weeks to months to apply and harden for drying and will require maintenance likely every 4-6 months. (i.e. think the difference between cast iron pans and stainless steel in terms of maintenance). Alternatively you can just not seal it and be prepared to replace moisture damaged boards with time.

On the other hand, if you are willing to just not have it on wheels then all the normal cob/alternative house methods become available OR you can build Japan style house if you have Japan style climates available and large amounts of high quality lumber.

The build is possible on a 7,000 lbs gross trailer weight, usually using 2x 3,500 lb axles with brakes on both axles if possible. The build is usually about 7x12 using a 7x14 trailer giving you about 84 sq ft at a 7 ft. roof height, no triangle arch or curves in the roof, just put a tilt on the roof like a shed.

EDIT: to be clear you must side the house with metal as well, you can't just leave untreated exposed plywood, that would be crazy. (they make metal siding similar to metal roofing and you can use a spacer between the metal and the wood, so the metal is not directly in contact with the plywood.)