The British Dental Association said for millions accessing NHS dentistry "is just a nice idea rather that a reality they can depend on".
During the General Election campaign, Labour pledged to create 700,000 additional urgent dental appointments a year, as well as recruiting new dentists to under-served areas.
The NHS GP Patient Survey found 51% of 3,546 respondents in the area covered by NHS Somerset Integrated Care Board were unsuccessful when they tried to book an appointment with an NHS dentist in the last two years.
Of those, 16% said no appointments were available, while 19% were told the dentist wasn't taking any new patients.
Louise Ansari, chief executive at Healthwatch England, said more fundamental reforms are needed, to give everyone a right to register with a local NHS dentist, in the same way as they can with a GP.
She said: "At the moment, dentists are not obliged to keep patients on permanently, which affects continuity of care, while the payment model for dentists does not incentivise them to offer fully preventative care to patients."
Across England, more than a quarter of people could not access an NHS dentist, which the British Dental Association said is around 5.6 million adults.
Areas with higher proportions of patients not being able to secure a dental check-up also saw higher levels of dissatisfaction with the services provided.
Of those who visited a dentist in Somerset in the last two years, 38% found the experience to be fairly or very poor.
Eddie Crouch, chair of the BDA, said "many have simply given up trying", with the union also estimating 5.4 million people no longer attempt to make appointments, as they do not believe they can get one.
In Somerset, 28% of 3,140 respondents who have not tried accessing dental care in the last two years said they had not think they could get an appointment.
Mr Crouch said: "We're still seeing sights like 'DIY' dentistry that belong in the Victorian era.
"None of these horrors are inevitable, and the new Government can turn the page."
He added: "Over a decade of underfunding and failed contracts has brought us here. A problem made in Westminster can be fixed in Westminster."
The figures also show 29% patients who have not tried to get an appointment in Somerset preferred to go to a private dentist, while 4% found NHS dentistry too expensive.
Dr Becks Fisher, director of research at the Nuffield Trust, said the survey results are "deeply troubling", and cautioned whether "Labour's pot of money" is sufficient to solve the situation.
"The new Government needs to build on the recent promising talks with dentists to deliver the reform of the contract they have promised – a contract which has failed dentists and patients for nearly twenty years," she added.
A Department of Health and Social Care spokesperson said the lack of dentist access is "unacceptable".
They added: "There are large parts of the country where NHS dentistry barely exists anymore.
"We will rebuild NHS dentistry, starting with an extra 700,000 emergency dentistry appointments. We will also reform the dental contract to encourage more dentists to offer NHS services to patients.
"Prevention is better than cure, so we will also introduce supervised tooth brushing for three to five-year-olds. These changes are fundamental to us building an NHS that is fit for the future."