Parcel courier DPD UK has said it is pausing its road delivery services into Europe, including Ireland, until at least Wednesday.
The courier has become the latest company to warn that British Prime Minister Boris Johnson's divorce settlement with the European Union had led to more complex processes at the border.
Seafood exporters said they have been hit by a "perfect storm" of bureaucracy, IT problems and confusion following Brexit.
Marks and Spencer said the new rules and regulations are set to "significantly impact" its overseas ventures in Ireland, the Czech Republic and France.
DPD said that up to 20% of parcels had incorrect or incomplete data, meaning they had to be returned to customers, and announced a pause to its road service into Europe and Ireland until Wednesday.
The company said in a statement: "The EU-UK Trade and Cooperation Agreement resulted in more complex processes, and additional customs data requirements for parcels destined for Europe.
"This, along with delays and congestion at UK ports for channel crossings, has placed extra pressure on our turnaround and transit times.
"We are seeing up to 20% of parcels with incorrect or incomplete data attached, resulting in these parcels needing to be returned to customers, so that the required data can be provided.
"In view of this unprecedented set of circumstances we believe that it is only right to pause and review our road service into Europe, including the Republic of Ireland. During this time, we will work with our customers to validate and correct the data we have in our system, to reduce the delays and enable us to resume normal service.
"This pause in our operation will be as short as possible and we intend to recommence this service on Wednesday January 13."
An Post has confirmed today that it is trading fully and smoothly with almost all of the UK online retailers for whom it delivers across Ireland.
In response to queries after DPD's decision, An Post said it has put in place a digital solution for all its UK retailers which simplifies the process and enables seamless despatch and immediate delivery.
95% of An Post's UK retailer customers have opted to use this digital system.
It said that while some delays should be expected due to UK retailers bedding in their new data requirements prior to shipping, the majority of items despatched through the An Post digital systems are released for immediate delivery on arrival into Ireland.
Donna Fordyce, chief executive at Seafood Scotland, said exporters faced "new bureaucratic non-tariff barriers" with no one body able to fix the situation.
"It's a perfect storm for Scottish seafood exporters. Weakened by Covid-19, and the closure of the French border before Christmas, the end of the Brexit transition period has unleashed layer upon layer of administrative problems, resulting in queues, border refusals and utter confusion," she said.
"IT problems in France meant consignments were diverted from Boulogne sur Mer to Dunkirk, which was unprepared as it wasn't supposed to be at the export front line. There have also been HMRC IT issues on the UK side that need to resolved ASAP regarding certification.
"A lack of knowledge and understanding of the required paperwork means some companies are ill-prepared for the new checks, which are taking far longer because of the mistakes being uncovered. When the systems settle down, checks should be carried out on samples from each load but now entire consignments are having to be checked to satisfy requirements."
M&S boss Steve Rowe warned that the trade agreement between the UK and the EU is causing problems with "potential tariffs on part of our range exported to the EU, together with very complex administrative processes".
Haulier industry body Logistics UK said deliveries were being delayed as lorries arrived in Belfast with incomplete paperwork.
In order to avoid a hard border on the island of Ireland, Mr Johnson agreed Northern Ireland would remain in the EU single market - but that has meant checks on goods arriving from Britain.
Labour has accused the British government of failing to properly prepare for the end of the transition period on 31 December.
Shadow transport secretary Jim McMahon said: "The Government promised it had a plan to make sure things ran smoothly for businesses and hauliers post-Brexit.
"It's clear the problems caused by its poor preparation and delaying tactics have not gone away. Ministers have to get a grip on this and make sure essential workers are actually able to do their jobs, or we risk seeing a repeat of the chaos on our roads at Christmas."
Meanwhile British Transport Secretary Grant Shapps has said "busy times" may return to UK borders as firms get to grips with new paperwork after the exit from the EU single market.
He told BBC Radio 4's Today programme: "The Cabinet Office is working very closely with businesses. It's not the case that goods are stopping flowing.
"I was studying, as I do by the hour at the moment, the flow at Kent and it's been picking up every single day of this year so far and you're seeing goods crossing the short straits and flowing perfectly smoothly."
Mr Schapps was pressed on whether pressures at the border will build because currently firms are avoiding it.
In response he said: "It's been a quieter start to the year, a lot of that's to do with anticipations, so the stockpiling that's going on."
He added that the chaos seen last month while the post-Brexit transition period was still in place was due to the French closing the border because of concerns over the UK strain of coronavirus and was "nothing to do with the change of paperwork".
"We may well see busy times again but actually at the moment the border is in fact flowing and it's flowing very smoothly," he said.