Online travel agent Lastminute.com failed to keep to a promise to refund people for Covid-hit holidays.
In December, it agreed to pay all outstanding package holiday refunds by the end of January.
But consumer group Which? said several customers still hadn't received a full refund after the deadline had passed.
"The refund process has been a very complex and difficult," Lastminute.com said.
Which? has called on the Competition and Markets Authority to uncover how many customers were not refunded in time and take appropriate action.
"Despite being given ample time to return all outstanding refunds to customers - as well as clear instructions regarding its liability for refunding both accommodation and flight costs - Lastminute.com has failed to meet its commitment to the regulator," said Rory Boland, editor of Which? Travel.
"The CMA must send a clear message that this kind of behaviour is unacceptable."
A CMA spokesperson said: "CMA action led to Lastminute.com committing to pay out over £7m to customers waiting for money back.
"They must now report to us on how they are complying with the commitments they signed up to and the deadlines agreed. Should it become clear that they have breached these undertakings we will consider further action."
Lastminute.com said: "Throughout this very challenging year, our customers have remained our number one priority, and we continue our commitment to dedicating our resources to helping them with their requests.
"Each customer request is unique, and often requiring a human touch-point and we've been working hard to get the money processed back through the system and into our customers' pockets as quickly as possible."
Sheryl McLeod booked a holiday for two adults and two children to Barcelona for July 2020 through Lastminute.com.
The firm told her in June the flights and hotel were cancelled and there were no alternatives, so the trip would be cancelled and refunded.
In September she was told by email that her refund was ready and she accepted the option of a cash refund.
For months afterwards Lastminute.com claimed it was finalising her refund. Then, on 27 January, she was finally sent £932.49 - more than £300 short of the £1,274.68 she was owed.
Claire Barder is another customer who told Which? she didn't receive a full refund for her cancelled holiday before the CMA's deadline.
Despite receiving confirmation of a refund for her July holiday to Barcelona, she was only given £431.75 - nearly £600 short as it did not include the flight portion of the trip.
After Which? approached Lastminute.com both customers were told they would receive their outstanding money back.
"We can confirm that the refund has been sent to the customers also for the flight," Lastminute.com said.
If your travel company cancels a package holiday for any reason, it has to give you a full refund by law within 14 days.
Similarly, if you booked a flight (which was departing or arriving in an EU country or the UK or on an EU or UK airline) through a holiday company and the flight was cancelled, you must be reimbursed.
But many people have been left waiting months for a pay-out during the pandemic as travel firms face a cashflow crisis.
Since the start of the Covid crisis, the CMA has written to more 100 package holiday firms to remind them of their obligations to comply with consumer protection law.
Last week it began an investigation of Teletext Holidays following complaints it has failed to refund people for trips that have been cancelled by the Covid-19 pandemic.
Some online travel agents have reported difficulties in securing refunds from airlines to pass on to their customers, meaning they have handed only partial refunds for cancelled package holidays.
But under the Package Travel and Linked Travel Regulations 2018, if a package holiday is cancelled by the provider, the customer is legally entitled to a full refund - including all travel and travel-related services - within 14 days.