In the years that I've owned a Win M94 Big Bore in .375Win I've never seen Winchester make ammo for it. Other ammo mfg's also list it, again I've never seen them actually make it. When they make it, Buffalo Bore .38-55 ammo can be shot in .375 Win rifles. See their page for the details:
Heavy 38-55 Winchester Field Proven rifle and gun ammo. Maximum firepower bullets. Best ammunition for rifles and guns for sale. Buffalo Bore. Strictly big bore. Strictly Business.
www.buffalobore.com
If you want to shoot it, reloading is going to be your best bet. Especially given the low power requirement of the 99. As reloading goes it's a fairly simple cartridge to reload. At low power the cases should last nearly forever, so you don't need a lot of them. And low power cases can easily be made from abundant .30-30 cases. I have occasionally seen these in stock:
https://eabco.com/375-winchester-brass-for-reloading-50/ They are made from unfired .30-30 brass.