I'm only aware of three photos of Vandervoort in Normandy, none of which are conclusive.
In the following photo, he has what could be a partly empty cloth rifle ammo bandoleer slung across his chest, which has gotten tangled around his waist. No weapon is visible, though.
And in this photo of a medal ceremony, while Vandervoort (on the far right) sports a pistol belt, has what appears to be a rifle with a M1907 leather sling slung over his right shoulder. It is not uncommon to see rifles being carried while the user wears a pistol belt, since they could utilize alternate means of carrying rifle ammo (such as cloth bandoleers).
A wider shot of the same medal ceremony isn't helpful, since Vandervoort's right shoulder is blocked by General Ridgeway, but note the soldier towards the right with a similarly slung M1 rifle with M1907 leather sling and pistol belt.
Looks like another clue comes from General James Gavin's autobiography "War and Peace in the Space Age". A footnote from the Wikipedia page for the M1 Carbine quotes the book as saying:
"Col. Gavin's love affair with his M1A1 carbine ended in Sicily, when his carbine and that of Maj. Vandervoort jammed repeatedly. Noticing that carbine fire rarely suppressed rifle fire from German infantry, he and Vandervoort traded with wounded soldiers for their M1 rifles and ammunition; Gavin carried an M1 rifle for the rest of the war."
I can imagine that after such an experience in Sicily, Vandervoort would have done likewise and most likely carried an M1 rifle from then on. So I would say that most likely the movie version is accurate, and it's certainly plausible.