Children aged 14 to 17 were sold vaping devices by 246 of the 634 retailers visited between January and March 2016.
The operation's report said compliance with rules prohibiting sales to under-18s was "disappointingly low".
The British Retail Consortium said major retailers had "rigorous policies and training" to ensure compliance.
E-cigarettes deliver a hit of addictive nicotine and emit water vapour to mimic the feeling and look of smoking. The vapour is considered potentially less harmful than cigarette smoke and is free of some damaging substances such as tar.
In October 2015, it became illegal for retailers to sell e-cigarettes to under-18s.
The operation, which was funded by the Department of Health, highlighted the wide variety of businesses supplying nicotine e-cigarettes and vaping liquids.
Most of the illegal sales (68%) were made at markets and car boot sales.
High levels of sales were also found at "other" premises, which included independent pharmacies, specialist e-cigarette shops and discount stores.
Fewest sales were made by national newsagents, petrol station kiosks and convenience stores, which are generally experienced in the sale of age restricted products and "specifically in the principle of challenging a young person about their age and asking for proof", the review said.
The operation found most of the vaping liquids were flavoured but some could be considered to be particularly "child appealing" - notably bubble gum, cherry cola and chocolate.
Trading Standards said the figures should be seen against the results from a 2014 drug survey, which showed that while one fifth of 11-15 year-old respondents had used electronic cigarettes, only 1% were regular users of the products.
Leon Livermore, chief executive at the Chartered Trading Standards Institute, said that while the results were disappointing it was important to consider them in context.
"More than 2.5 million adults use electronic cigarettes and evidence suggests the products are now the most popular quitting aid for smokers in England," he said.
"And these products are being sold in a wide variety of retailers and many of them will have little or no experience of challenging age restricted sales."
He said that where an illegal sale was made, further advice and guidance were given to help the business achieve compliance, but that penalties for the offence can be a fine of up to Â£2,500.
Nicola Blackwood, the public health minister, said there was strong support for restrictions from businesses when they were consulted, and added: "As the school holidays are upon us, this is a timely reminder of their obligations under these regulations not to sell nicotine products to under 18 year olds."
Commenting on the results of the operation, the Electronic Cigarette Industry Trade Association said it welcomed the introduction of legislation to prohibit sales to under-18s and it was "disappointing" that compliance was "so low".
Chief Scientific Officer Tom Pruen said: "While e-cigs offer the potential to save many lives by reducing the harm caused by smoking, it is important that they do not form a gateway into nicotine addiction for people, especially teenagers, who do not already use nicotine.
"Fortunately, while teenagers appear to be experimenting with vaping, this does not seem to be transferring into regular use, unlike with smoking.
"However, this is a genuine concern and one which requires careful monitoring, and responsible action from the industry at all levels."
The British Retail Consortium said: "The sale of e-cigarettes and vaping liquids is regulated by law and is subject to the same levels of control as tobacco products.
"Every major retailer has rigorous policies and training practices, including awareness campaigns and till prompts, to ensure that the sale of such products is in full compliance with the law."
Almost 40% of sellers targeted in an operation by Trading Standards in England were caught illegally allowing under-18s to buy e-cigarette products.