Several units had them in Normandy as they appear on the unit returns for 1st and 8th June 1944.
21st Panzer Division hadsome Panzer III M in Normandy.
4 of them in the HQs of Panzer-Regiment 22 and its 1st Abt. along with 2 Panzer III Befehlswagen.
116th reported 11 of various models, Panzer-Ersatz und Ausbildungs Abt 300 had one. While 9th SS, 10th SS and 12th SS all report Panzer III Befehlswagens.
In total around 30 Panzer IIIs took part in the Normandy campaign.