The Alps during winter aren't as bad as one can imagine, as far as supplies go. Cold meant that storing goods was possible with minimal risk of spoiling. Of course, food stuff had to be brought to the front line, often carried to mountain tops, where positions were frequently devoid of anything remotely resembling a road. Paths were cut for this purpose, where possible, or vie ferrate on the alpine rocks (a couple of pictures here), or pulleys to lift heavier stuff. Of course, men were aided by beasts wherever beasts could help - donkeys and mules and horses served in the war as well. There is a large number of videos (one and two) that can give you a better idea of the sort of equipment available and the conditions of the front.

One must remember though that, generally, higher also meant fewer men to supply so that provisions could be brought far less frequently. Solitute was the bane of the men high on the top; going weeks without a letter or package. But meat could be stored in chunks, right there in the rock flanks, or even in the ice mass.

Where more men were to be supplied, larger amounts of food stuff, ammunitions, and all sort of things had to be stored. If you look at the shape of the Italian front you can imagine the supply bottleneck right in the middle: the railroad to Udine (where the High Command and main supply center for the front was located) was a crucial point, with hundreds of thousands of men, both military and civilians, living in the rear of the active army. To ensure its functionality, the High Command had pulled materials, coal, and wagons from the rest of the peninsula, to the point where many non strategic lines had to cut the number of trains per day to half or less (despite the state of war, a national rationing of coal due to largely reduced imports had brought consumption from 10 million tons to 8 in 1916 and 5 in 1917).

As you can see, going to the top of the mountains might have been hard for the men doing the job, but as far as logistics went, it wasn't the hardest part. The production of the entire peninsula had to be moved along towards the extreme north-east. And the production was scarce - Italy at war found itself in need of everything.

Food was lacking. The Ministry of War had reduced rations to barely over 2,000 calories per day during the last months of 1916. The situation had improved slightly with after Caporetto (the supply line and general defensive position also meant that food arrived for real and more or less on schedule - something that wasn't always true before) with daily allowance reaching around 2,800 kcal (the physicians of the ministry argued that the smaller Italian soldiers needed 2,500 calories compared to the 3,200 of a British). Those consisted fore the most part of bread and meat. A front line soldier had 350 grams of meat per day; one in the rear 250. For the reserve meat was granted only three times a week but once a week stockfish was given. With the US entrance to war Italy had secured some supply of canned meat that was served as well.

By that time, the Ministry had to count carefully the numbers of animals like cows, pigs, even horses and donkeys for slaughter. Animals were needed for work and reproductions - and the impact of seizing live animals would have been heavy on the agricultural sector that, if possible, suffered even more than the breeding one. To provide the soldiers with the roughly 600 grams of bread per day, Italy needed to import grains - imports during the war made up over a third of national consumption staying around 20 million quintals from 1915 to 1919 (consumption according to the Minsitry of supplies had been 53 in 1914, rising to 59 in 1915 and 64 in 1916 with an estimate of 68 in 1917). To add to the lacking harvest (23 millions for 1918) the Ministry had tried to rely on other sources: barley, oat, rye (together around 8 millions) and 41 millions from various legumes (accounted already and harder to process into food stuff for the army), 3.4 millions of rice (0.7 went straight to army consumption). The Ministry was also worried about sugar: rising demand, lower production - Italy was short half a million quintals from esimated needs. I'd love to tell you that oil and cheese and butter were enough; but it would be a lie. Butter was short 53,000 quintals (25% of pre war demands) and cheese short 400,000 quintals (same quota of demands); I have no eact data on olive oil, but it wasn't enough.

And that's ignoring the other stuff men needed, such as clothes. Wool was something Italy produced in decent amount, but grade wool (for winter clothes) was scarce and subject to rationing and destined to the Army. Other fibers, such as cotton - necessary not only for clothes but also for ropes and especially balck powder - was not produced in Italy in any significant amount. It had to be imported (from the States) and replaced wherever possible with hemp. Linen used to be imported from Russia but in 1917 the imports ceased entirely.

If you are wondering why didn't they increase production, you must remember that - especially in early XX century Italy - production relied on men and beasts alike. And men and beasts were needed to fight the war; those who weren't on the front working in war factories. Therefore production of foodstuff dropped almost in every sector. Imports also dropped to some extent and Italy had to negotiate carefully with the allies for what it needed.

Summing up, while the logistics of mountain war certainly caused a lot of trouble, those could be overcome with the thing the Italian Army did not lack: labor force. The real issue was the original bottleneck: lacking resources, lacking supplies and infrastructure, that made providing the men on the front a harder task for the whole country.

I have taken the numbers from the contemporary relations of the Ministries of War, Ammunitions and Supplies. I haven't checked every single number though and if you believe something's off, let me know and I'll check them.

For additonal sources: P. Melograni - Storia politica della grande guerra

Edit: spelling