Thanks for all the replies guys! Someone on Armslist is selling a box of 20 for $90... I'm sure he won't sell it at that price! The gun auction sites aren't much better. Midwayusa sells it for $45 a box when they have it in stock, but its unavailable at this time. I check the major big box stores almost every day, like Gander, Cabelas and Bass Pro. The problem is that Winchester only makes it on a limited seasonal run, so my only hopes of getting some factory ammo this time of year might be some new old stock collecting dust at a gun shop where it's not such a "popular" caliber. My gun shop had 3 older white boxes about 2 years ago but I didn't have a 375 Winchester at the time. Needless to say, someone came in a bought all three boxes over a year ago.
I ordered bullets and reloading dies online already. I found some 38-55 brass at Midwayusa that I can re-size with my 375Win dies, then trim to the correct length. The brass is a little thinner than 375 brass so I will have to load them on the conservative side, which I typically do anyway. Buying a box or two of factory ammo just gives me the correct brass that I can use for many years.
Tomorrow I'll call these local guns shops you mentioned. I don't mind a little road trip for some hard-to-find ammo. I don't plan on hunting with the rifle this year because I have several other guns already sighted in. I also didn't know that ammo can be shipped to our house again, so thanks for that update! I wonder how many web sites know that.
Thanks again for the information!