The film released by the charity showed the animals being beaten and, sometimes, deprived of food and water for long periods.
The Shropshire Star has made several attempts to contact the owner of the North Shropshire business, but has been unable to get any response.
Some of the calves involved are understood to have come from farms that supply dairy giant Muller – which in turn supplies supermarket chain Sainsbury's.
Muller had already said it had told its suppliers to cease using Oaklands Livestock Centre, with a spokesman saying: "In light of this evidence of mistreatment, we are therefore instructing farmers to cease supplying Oaklands with immediate effect."
Now Sainsbury's has also voiced its concern about the reports, and said it would be looking into the matter.
A spokesman for the supermarket company said: "We have processes in place to prevent these unacceptable practices and we are urgently investigating."
The calves are understood to have been collected by the livestock centre from farms, to be taken for slaughter.
Footage filmed during the investigation at the centre in Weston-Under-Redcastle, near Wem, showed workers kicking and pushing calves down ramps out of the back of trailers, with another worker seen appearing to punch a calf.
The Animal Justice Project says it captured the footage during an undercover investigation between November last year and March.
Edie Bowles, Advocates for Animals solicitor, said: “The footage gathered by Animal Justice Project has called into question the compliance of several welfare laws that farmers are required to follow.
“It is essential these breaches are thoroughly investigated and appropriate action taken. The public needs to have confidence the laws passed to protect farmed animals do not exist on paper only.”