A formal complaint has been lodged against MP Jared O'Mara claiming he has breached parliamentary standards.
In a letter to the Parliamentary Commissioner for Standards, a constituent alleges the MP has failed to act in the public interest and has used public money for his own benefit.
Jodi Garth, who brought the complaint, has called for a full investigation into the Sheffield Hallam MP's affairs.
The Parliamentary Commissioner for Standards has declined to comment.
Ms Garth believes that by delaying his resignation and continuing to receive a salary Mr O'Mara presents a "clear case of an MP giving preference to their personal interest over the public interest".
She points to the fact Mr O'Mara did not take part in any of the key parliamentary votes held since postponing his resignation.
She told the BBC: "I feel he's not properly fulfilling his duties [and] I really want to know what has changed as he's not come out to say why he's withdrawn or postponed his resignation; it just leads me to think it's for the wrong reasons."
Mr O'Mara, who has been an independent MP since leaving the Labour Party in 2018, announced his intention to resign in July, saying he would step down in September following Parliament's summer recess.
At the time he said he was taking time out from his official duties to deal with "mental health and personal issues", adding: "I am not in any fit state to continue and nor would that be appropriate if I was."
According to Hansard Mr O'Mara last spoke in parliament in October 2018 and has not voted since April, while his constituency office has been shut since August.
Analysis by Liz Roberts, political reporter BBC Radio Sheffield
This is certainly an unusual scenario.
An MP with no party, no offices, no staff and who's too unwell to perform the job.
It's a set of circumstances which, when combined, leaves constituents with no representation whatsoever and no-one obvious to turn to.
There seems to be no precedent which would help indicate who should take responsibility.
That's clear from the number of different departments at Westminster I've been passed around since trying to find answers.
Mr O'Mara says he thinks he'll be ready to stand down by the end of next month once the "administrative process" of winding down is complete.
If he doesn't, it's likely a looming general election will make the decision for him, whether the commissioner decides to investigate or not.
Ms Garth said she had felt "shocked and disappointed" at the lack of recourse for constituents when their MP ceases to engage in their duties.
Liberal Democrat chief whip Alistair Carmichael MP said it was " unacceptable" that constituents should be left without representation as a result of illness.
"Mr O'Mara is not the first MP to become ill and it should not be beyond the wit of man to find a way of providing a service to his constituents while allowing for him to recover," he said.
When approached by the BBC for a comment, Mr O'Mara said he was "still leaving" and was in the "administrative process of winding everything with IPSA (Independent Parliamentary Standards Authority)".
He said he expected the process to be concluded by the beginning of November "at which point I will contact the Treasury to conclude matters".