Kevin Thomas, was so concerned when he didn't receive any post, he approached staff at the Wakefield sorting office on Denby Dale Road on Wednesday.
Mr Thomas says staff told him the system was overwhelmed with sickness and holiday leave, as well as a new system that is currently broken.
He said: "I was told they have a backlog of over 1,000 trays of unsorted post from the last few weeks and no way to process it."
Mr Thomas also says he was told that only recorded and tracked post was being handled with everything else unsorted and undelivered.
He added: "I spoke to Royal Mail's official help line and they say there is "no issue."
"This is a disgusting letdown of public trust.
"People have medications delivered by post.
"People will be missing important appointment
."I spoke to one woman who would be missing running the Manchester Marathon due to this, another on a deadline for applying for the right to stay in the UK because her documents have gone missing."
Debbie Pottage from Lupset said she had been made aware that the problem had been going on for a month.
She said: "People are missing important mail with hospital appointments arriving with very little time to make plans."
Simon Skinner of College Grove said the whole of the WF1 postcode appeared to be having problems.
He said: "We've not been getting regular letters for about two weeks.
"I went down to the sorting office to ask them to explain the situation and was told the system was being restructured and that 40 members of staff were of sick with Covid.
"I picked up 15 letters addressed to me that had not been delivered."
Mr Skinner was told it could be another fortnight before it was resolved.
He added: "Our postman is still coming round but he is delivering fewer letters and more parcels."
A Royal Mail spokesperson said: “The health and safety of our colleagues and our customers is our number one priority.
"In the Wakefield area, we are experiencing some disruption to service due to COVID related and other sickness absence.
"We aim to deliver to all addresses we have mail for, six days a week.
"If resourcing issues, associated self-isolation and safety measures prevent this, we’ll deliver at least every other day.
"We apologise to customers for any inconvenience they may experience as we work hard to resolve the situation.”