Virtually all soaps, shampoos and toothpastes contain ingredients derived from corn, soy, and/or coconut. For someone allergic to all three, or with multiple chemical sensitivities (MCS) the options are extremely limited.
Here is a way to get your hair and teeth clean with good old baking soda. This is about as hypoallergenic as you can get.
Take about 2 tablespoons of baking soda (bicarbonate of soda) and add enough water to make a paste. Rub into your scalp and rinse.
If desired, follow with an acid rinse to restore PH balance: Use about 2 tablespoons fresh-squeezed lemon juice or apple cider vinegar mixed with about a cup of water, being sure to get the ends of your hair.
Rinse with plain water.
You can use plain baking soda or a baking soda/additive-free salt mixture as toothpaste.