Officers confirmed they received reports of criminal damage from 3 to 4 July at Crookes Cemetery.
Mrs Grayson said that while the cemetery management were “very kind”, the onus was on the grave owner to repair any damage.
“Putting things right will come later on – we just want to catch who did it,” she explained.
She hopes any witnesses will have the “courage and conscience” to come forward and help the police so that the cemetery can be restored to a “peaceful and calm place”.
South Yorkshire Police said: “Inquiries are ongoing and if anyone has any information, please get in touch.”
The BBC has also contacted Sheffield City Council for comment.