Thomas Cook customers who had booked holidays with the firm have submitted 60,000 refund forms in the first hours of operation since a special website for the process was launched.
The Civil Aviation Authority said the process was now running smoothly after initial delays caused by high demand.
Customers had complained they had tried to submit the claim form several times, but kept receiving error messages.
The CAA blamed "unprecedented demand" and urged users to try later.
In total, the aviation regulator has to refund some 360,000 customers.
It will take 60 days for people to get their money back, the CAA said.
"We would like to thank Thomas Cook customers for their patience during the peak claims period earlier today," the CAA said in a tweet.
We have now had 40,000 claims submitted through our online form.— UK Civil Aviation Authority (@UK_CAA) October 7, 2019
We would like to thank Thomas Cook customers for their patience during the peak claims period earlier today.
You can find advice and the claims forms on our website:https://t.co/7Rsubnbg5g pic.twitter.com/Mw8uKoJ3NB
Several people told the BBC earlier on Monday that they had completed the form, but when they tried to submit it they received a message telling them "an unhandled fault occurred while processing this flow. Please contact your administrator".
Joly Shapley told the BBC he had tried to put in his claim at 08:16 BST, but with no success.
"At 8:20 am I called their helpline and after 10 minutes I got through to a call handler who... suggested I try later. She assured me [there was] no problem making multiple attempts but told me that she was not allowed to take my claim over the phone.
"Since then I have made seven further attempts to complete this online form. Sadly, although the CAA had extra time to prepare this process it appears to be too fragile for its purpose," he said.
Sue Nicolson said she had tried to submit her claim a dozen times.
"The 60-day timescale for refunds only starts once they have received the claim, so how much longer are we going to have to wait for the thousands of pounds we are owed for a holiday we were supposed to depart for this Friday?," she said.
The CAA said people who had paid by direct debit would get their money back by 14 October.
Anyone who bought a package holiday with Thomas Cook will be covered by the Air Travel Organiser's Licence scheme (Atol). Customers would have received an Atol certificate when they booked. This means the cost of any holiday booked with the collapsed firm will be refunded.
The CAA launched the refund website as the final flight bringing holidaymakers back by emergency repatriation landed on Monday morning.
The few remaining passengers who did not return on a CAA-organised flight will have to make their own plans, although those covered by the Atol scheme will be refunded.
CAA chair Dame Deirdre Hutton said she was "deeply relieved" that "Operation Matterhorn", the two-week operation to return 150,000 passengers to the UK after the package tour company collapsed last month, was over.
"Staff worked like Trojans 24 hours a day to help everyone, but that was only task one, now it's task two," she said, referring to the refund process.
Staff left without pay
Meanwhile, staff of the collapsed firm have not been paid for September and have to apply for their salary and redundancy related payments to the Insolvency Service's Redundancy Payment Service (RPS).
About 9,000 staff in the UK were left jobless when the business failed to secure a last-ditch rescue deal.
The travel firm collapsed in the early hours of 23 September, after failing to obtain rescue funds from its banks.
An inquiry has been launched by the Business, Energy and Industrial Strategy Committee, with MPs focussing on the directors' stewardship of the company.
The Financial Reporting Council, the accounting watchdog, will also investigate the auditing of the company.