Cllr Noreen New (Lib Dem, Paddox) urged the rethink to help those unable to book slots particularly as other Covid restrictions had been lifted across the country.
Her notice of motion considered at this week’s (April 26) full council meeting of Rugby Borough Council.
Cllr New said: “I totally understand that the booking system has been working very well and it does eliminate parking, traffic and congestion on the roads leading up to Hunters Lane site.
“However, there are a lot of people who are not computer literate, who do not have smartphones, who are unable to access the booking systems.
But the suggestion failed to find support with both the Labour and Conservative groups.
Cllr Peter Eccleson (Con, Dunsmore) said: "This council has declared a climate emergency and queuing traffic, as you acknowledge, will increase the amount of pollution.
“As a user of the site on a regular basis I have found the appointment system very good and the majority of people I have spoken to find it very good.
Cllr Maggie O’Rourke (Lab, Benn) added: “It is in our ward in the urban area and we all know what it was like before Covid with people queuing up all down the street.
“We have industrial action going on at the moment so probably having access to the booking system and getting them in and out is something that we want to stick to because more people are going to be using it.”