The DVLA's new online system to check driving licences following the abolition of the paper counterpart crashed on the day of its launch yesterday.
Users of the new View Driving Licence service complained that the system was not working properly yesterday morning, says the BVRLA.
Several took to Twitter to complain, while the BVRLA had feedback from a number of members that there have been delays in rental branches.
The DVLA says that these sorts of delays are to be expected on the first day of a new system during peak periods of demand.
Gerry Keaney, chief executive of the BVRLA, said: "Replacing paper forms with digital services is a great idea, but the Government has gone about this the wrong way by rushing the process and not giving enough warning to motorists.
"The online system being offered by the DVLA is far from ideal and the car rental industry is working with it as best it can. We are confident that our members will keep their GB licence holder customers up-to-date with any new procedures, which should minimise any disruption for travellers."
“Customers can take extra precautions by ensuring that their rental company has access to their endorsement information – available via the DVLA’s Share Driving Licence service or its call centre.”
The AA is among the companies to say its members are encountering issues, while a car hire firm in Bristol told the BBC it had a queue of 20 people waiting as staff were unable to access their details online.
DVLA chief executive Oliver Morley yesterday told the Independent: "We are currently experiencing exceptionally high demand for this service.
"We are aware some customers are experiencing issues with the website and we are working very hard to resolve this as quickly as possible. We are sorry for any inconvenience."