A petition has now been launched and dozens of people have made their views known on the town's Facebook networking site.
Many are calling for a new supermarket to provide competition for Budgens which has a monopoly in Rye and closes its doors on Sunday.
A number of Rye councillors are backing the calls with Deputy Mayor John Breeds commenting that he gets more complaints about the need for better shops in Rye than any other issue.
The problem has been made worse with the closure of Woolworths and there is concern that the building, in the heart of Rye's retail area, now looks set to become a public library.
Rye's Facebook site says it wants to gather views to influence the Marketing Group and Rye Chamber of Commerce.
Many have commented that trade is leaking out of Rye to nearby Tenterden, across the border in Kent, particularly on Sundays.
A petition for a supermarket that opens late and on Sundays has been launched by Lucy Emma Richardson at Clouds Emporium.
She said: "We need this. It would bring life back to Rye. It will not spoil Rye's uniqueness just bring back a much needed economic boost to the town. I and everyone I know travels to Tenterden or Hastings to get our shopping.
Wendy Harvey said: "It would be better for the whole town if more shops are open seven days a week, and we badly need another supermarket - a small Sainsburys or Tesco or Co-op would be great.
"Rye needs reasons for people to want to come to the town as well as the quaintness and all the tearooms and antique shops.
Amanda Bheir commented: "Rye is dying - please, please, please can we have a supermarket that opens Sundays - it will help re-generate the town.
Fiona Hooker said: "Shops should be open every day especially a supermarket. Competition would be good.
"You can see people's faces coming off the train , especially in the summer, when they see the supermarket shut."
Debbie Cooper said: "Rye must be the only place left in the world that shuts up shop on a Sunday. It's about time the people of Rye had their say."
Jamie Haffenden remarked: "I spent the first 19 years of my life in Rye, yet when I visit now I struggle to feel anything homely about it - it does feel like its dying and its a real shame.
Mary Faulkner said: "I cannot believe that a tourist town is closed on a Sunday a decade into the 21st century. My family lived there from the 1500s. None of us are there no there now, I wonder why!"
Emily Plummer said: "It's ridiculous for a small independent supermarket which is meant to serve the local community to not open on a Sunday, which would not be a problem if there was somewhere else to shop.
"It's about time seaside towns actually remembered that it's the locals who are their bread and butter during the winter. People would use the shops in Rye more if they were open more often.
Clare Murray commented: "It's ridiculous that in a tourist town like Rye you can't just pop out and buy a pint of milk on a Sunday - pity the poor weekend holiday lets – they don't know what they're in for.
Stephen Jempson said: "Jempsons has traded in the local community for 75 years in June this year. From its inception in 1935 the business has never traded on Sunday.
"Despite this, the company has prospered from a small village bakery into a group comprising four supermarkets, a petrol station and five cafes.
"We are often asked why we don't open on a Sunday. We receive many letters and calls on this some of which are for us and others against. We do attempt at all times to try and outline our views clearly and politely, all the while respecting others' views on the subject.
"It is disappointing and we are saddened by the attitude, but it is factual to state we have received correspondence from a Town Councillor stating they will never set foot in our stores again until we alter our position.
"There are a number of reasons why Jempsons believe in not trading on a Sunday.
"The freedom to shop takes away the freedom of someone else who has to work. And it's not just the shopworkers - police, security, banking, distribution, transport, health and catering staff all have to work when shops are open.
"Many people might only want to use the shops for an hour or two, but these workers are already spending six hours away from their family and friends. It's not much use giving them a different day off in the week either when their children are at school and their friends are at work.
"Sunday trading has a direct effect on a local community. Not only does it have an impact in terms of noise and congestion, it also hits community activities. For some that might be getting together with others at church meetings, for others it might be playing football in the local league.
"Are there really those that believe that big stores want longer opening hours on Sundays just because they want to make their customers happy? The real motivation is to make more money.
"In the last decade, 30,000 small shops have had to close because they can't keep up with the big stores. On average, that's over eight shops shutting up for good every single day.
"For many people in this country, Sunday has a particular religious significance as a day set aside for worship and a day that's different from the rest of the week - a view that's not shared by everyone but one that we should respect as it has served the country well for many years.
"In 2005, 3.6 million of us in the UK regularly worked more than 48 hours a week, and the incidence of long hours working in this country is more than two and half times the average for the EU. It's perhaps not surprising that work-related stress affects more than one in five people and is the cause of more than 13 million lost working days a year. What we need is time to stop and time to rest.
"87% of people think it is important for family stability and community life to have a common day off each week. (NOP 2005)"