We were there in September and had a Walmart grocery order. We had a few fridge items in the order, Walmart delivered to bell services and held it for the time we were in Trader Sam's (probably about 1.5 hours after delivery before it was brought to our room). They never mentioned not refrigerating cold items, I'm pretty sure ours were still cold so I assume they put them in there.
We've always had good luck with the on-site hotels for delivery. Off-site tends to be iffy, but you should have no problem at the DLH.