The petition, launched by Telford & Wrekin Council leader Shaun Davies, is demanding the government intervene over the controversial proposals.
The Government has resisted the call to step in, and responded by saying “it is in the best interests of the local NHS and patients for decisions such as this one to be made locally”.
If the petition reaches 100,000 signatures it will be considered for debate in parliament.
It comes as 15 union leaders across Shropshire and the Midlands have signed a letter calling on Health Secretary Matt Hancock to step in over plans, and to sanction the immediate removal of hospital boss Simon Wright.
Union bosses say closing A&E at Princess Royal Hospital overnight will put lives at risk.
The letter from the Shropshire & Telford Trades Union Council has also been sent to Telford MP Lucy Allan, Wrekin MP Mark Pritchard, Ludlow MP Philip Dunne, North Shropshire MP Owen Paterson and Daniel Kawczynski, MP for Shrewsbury and Atcham.
It states: “As representatives of trade unionists who are working within the health service and also those who use, rely on and love our NHS, we are appalled by Shrewsbury and Telford Hospital NHS Trust’s decision to close the PRH A&E department overnight from December.
“This puts the lives of our members and your constituents and their families at risk.
“We therefore demand that you call upon the Secretary of State for Health to intervene to ensure that this grave decision is reversed.
“The people of Shropshire and Telford no longer have confidence in the ability of Simon Wright as CEO of SaTH to protect the health and well-being of our community and we therefore call on you to demand his immediate removal from that role.
“Finally, you have a duty as elected representatives of your constituents to demand that your government sufficiently funds the NHS to meet the needs of the people of Shropshire and Telford.”
The planned overnight closure of Telford's A&E is now due to happen from December 5, between 10pm and 8am, if additional staff cannot be found.
The trust says the decision has been taken due to staff shortages, with major gaps in consultants and middle-grade doctors leaving no alternative.
Trades Council president Andy Brown said: “Our NHS has a special place in the hearts of working people; we rely on it to protect us and our families and therefore the counties MPs need to be taken to task over their Government’s disastrous record on the NHS.
“Union members in the county admire the skills and dedication of those that work in the NHS and we stand with them to ensure that the NHS survives and can meet the needs of our communities."
Mr Wright, chief executive of the trust, said no final decision has been made to suspend overnight emergency services at PRH and "every conceivable effort" is being made to prevent any such decision having to be made.
He said: “We absolutely do not want to make this suspend overnight emergency services at PRH; however it would be irresponsible for us not to have robust plans in place to ensure the continued safe and dignified treatment of our patients. To take such action would likely be the biggest decision anyone on the board would ever have to make.
“The A&E department at PRH currently remains open 24-hours-a-day, seven-days-a-week, however if the recruitment of additional staff is not possible the board have agreed that an overnight closure will be unavoidable because of the impact staff shortages are having on patient safety.
“The decision on whether to suspend A&E services overnight at PRH will be made at the next trust board meeting but prior to that our clinicians will be conducting a major desktop exercise to test the pathways for every patient to make sure they are safe. We won’t consider changing the service until we are completely assured that watertight plans are in place for every patient to receive the safest pathway of care.
“In the meantime we will continue to look at ways of avoiding the potential overnight closure. Clinicians at SaTH believe that both A&E departments could be maintained in the short-term if the trust had an absolute minimum of seven additional middle grade doctors – taking staffing levels up from 11 to 18 – and an absolute minimum of 15 more registered A&E nurses.
“The trust needs stability to ensure the improvements we are making are continued and embedded – which is in the best interest of our patients.”
Consultants at SaTH recently issued an open letter to the people of Shropshire, Telford & Wrekin and Mid Wales stating their support for the trust’s leadership.
It read: “We have a clear vision for the future of our hospitals. This year, we have appointed many new consultants to the trust who want to make their home here and help us realise this vision.
“We will continue to work and support our managers and members of the board to bring this to fruition. Safe and effective care in Telford, Wrekin, Mid Wales and Shropshire requires stable leadership; any major change to the leadership team would be a misguided and unnecessary distraction at this key time.”
SaTH's board will be required to provide the final sign off on the temporary closure plan.
Campaigners are also expected to join a protest march through Wellington on Sunday, organised by the campaign group NHS4All and Telford & Wrekin Council.