At least one customer was able to see another person's private information on the Web site, which launched on Monday, and many more customers trying to log on to the Nectar site failed to gain access.
In a statement on the Nectar site, Loyalty Management UK, the company that runs the service, said: "We are currently experiencing very high volumes of traffic on the Nectar website. As a result you may experience difficulty in accessing our site."
Loyalty Management UK recommended that customers wishing to register should call the Nectar helpline to request a paper form to complete and post in.
The problems at Nectar follow other first-day failures of high-profile Internet sites. In January, the 1901 Census Online service crashed after more than one million users tried to access the site in the space of an hour. The site is still operating as a test site.