Sandalwood, neem, turmeric and glycerine soaps, including the classics that have been made the same way for decades.
Available in the United States. No customs at your door.
The reason to import soap is specificity. Mysore Sandal has been made by the same state-owned works since 1916 and smells of actual sandalwood oil rather than a synthetic approximation. Neem and turmeric bars from Ayurvedic makers use meaningful quantities of both. These are not exotic in India and are genuinely hard to buy elsewhere.
Soap is heavy for its price, which makes it one of the better things to add to an order rather than the reason for one. A single bar rarely justifies international freight; six of them alongside something else usually does.
If you have sensitive skin, read the ingredient list rather than trusting the word natural. Essential oils at the concentrations used in strongly fragranced bars are a common irritant, and patch testing a new soap is worth the two days.