The test cases were highlighted by Irish national broadcaster RTE on its main current affairs programme Prime Time.
At outlets in Swords, County Dublin, the two supermarket chains accepted online orders and payment by debit card, while the off-licences took cash on delivery. None of the businesses asked the age or identity of those ordering.
The exposure has prompted one retail group, the 1,500-member Convenience Stores' and Newsagents' Association, to call for online alcohol sales to be banned. Spokesman Vincent Jennings said that while "the vast majority of retailers" had a comprehensive policy on ID cards at shop level, that did not apply to online sales.
Government action was needed to stop alcohol being available online, he added. "We have to accept the product for what it is either it's a controllable substance, or it's not."
Both supermarkets accepted responsibility and promised remedial action. Tesco said that online transactions represented "less than 2%" of alcohol sales, and added: "Tesco does not condone the delivery of alcohol to anyone below the legal age limit. We are investigating the issue and will take all necessary and appropriate action."