The incident happened at Montgomery Hall, Wath-upon-Dearne, where more than 8,000 people have had the Pfizer vaccine.
MP for Wentworth and Dearne, John Healey, defended staff at the venue after the blunder came to light.
He said: “I want to speak up for the incredible effort by our NHS staff and volunteers - we’re well on track to vaccinate the top four national priority groups by the February 15 deadline.
“More than 8,000 people have had their jabs at Montgomery Hall and not a single appointment has been cancelled - despite flooding in the area and two days of bad snow.
“I’ve been at Montgomery Hall and volunteered at other centres in the area, and seen first-hand the commitment of NHS staff and others delivering the vaccine programme, working seven days a week on top of their day jobs.
“I have nothing but praise for what they’re doing.”