The Ukraine Sponsorship Scheme, also known as Homes for Ukraine, allows citizens to volunteer to house refugees fleeing the war in Ukraine.
Sponsors offer accommodation for at least six months, with those receiving sponsorship allowed to live, work and study in the UK for up to three years.
The Refugee Council said it had received reports of delays at every stage of the application process, and of necessary documents not reaching refugees quick enough for them to travel.
Head of advocacy at the charity, Andy Hewett, called the scheme "unfit for purpose".
He said: “Responding to a serious humanitarian crisis by offering complex visa routes, putting paperwork and bureaucracy before people was always going to have tragic consequences."
Some families are also not having their applications processed together, creating difficulties when not all visas are granted at the same time.
Others have flagged the potential for safeguarding issues from a lack of thorough checks on prospective sponsors, and there have been national reports of refugees becoming homeless after being turfed out by their hosts.
Refugees have also been arriving through the Ukraine Family Scheme, which allows Ukrainian nationals to join family members in the UK.
Across the UK, 107,400 visas had been issued to Ukrainian refugees through both schemes as of Tuesday, with 53,800 arriving in the UK by Monday.
The Government warned the data is likely to contain some duplicate records, while around 2% of sponsors could not be assigned to a local area.
The number of successful applications has increased compared to last month. As of April 19, 71,800 applications had been granted through both schemes – including 69 in High Peak – and 21,600 had arrived in the UK as of the day before.
The Home Office said some refugees are choosing to stay put or travel elsewhere, which may explain some of the gap between arrivals and the number of visas granted.
A spokesperson called it "one of the fastest and biggest visa schemes in UK history" adding they are now “processing visas as fast as quickly as they come in” as a result of changes to the application system.