I tried to buy something at a duty-free shop but didn’t have an Immigration stamp in my passport, so they wouldn’t sell it to me duty-free.

Only non-residents (people with no domicile in Japan) are eligible to purchase duty-free goods, and certain criteria must be met. For this reason, you will be asked at a duty-free shop to show your passport to confirm that you meet the criteria for duty-free purchasing. If you go through an automated gate at airport immigration upon arrival, your passport will not get stamped, so a duty-free shop will not be able to determine how long you have been in Japan, and they may not be able to confirm your eligibility.

If you plan to make any purchases at a duty-free shop, be sure to get your passport stamped upon arriving in the country.

Note: An automated gate is for verifying one’s identity using one’s passport and fingerprints as part of a system that enables automation of the arrival and departure procedures into and out of Japan.