I believe they can, as the interesting thing about Sardinia is it has only one major port, Cagliari, all the way down in the extreme south of the island, Since the British are going to land there anyway, and there are no German forces on site, once the port is secured, the Italians have only two minor ports, Oristano and Olbia, with which the Italians can draw supply.
Note-Until the British can repair the damage done to Cagliari's port facilities the Italians will be able to reinforce Sardinia and supply there on-island forces. But there are no rails on Sardinia worthy of the name, meaning only the port of Oristano is in supply range of contesting British operations in and around Cagliari.
A large wide valley running between Cagliari and Oristano allows easy movement and local supply for the Italians, but the reverse is true for the British once Cagliari is operational as a major port again. That would, depending on circumstances, depend on how long it takes to finish repair efforts. 4-10 weeks? Once Cagliari is up and running, air superiority can be fully established over the rest of the northwestern Med, meaning the RAF can suppress the ports of Oristano and Olbia, effectively shutting down the Sardinian Campaign for good.
Olbia is behind a large amount of very heavy rough terrain, isolating it both from threatening the British operations on most of the island but also protecting the port from being taken by the British in any real fast time frame.(1)
1) This brings it all back to the Germans. Does Hitler throw a tantrum and throw in the Luftwaffe? Even use Italian shipping to feed in German Army units to Sardinia? Only the OP knows whether the Italian merchant marine and navy even has the ability to project power now. But I would imagine they WOULD be able to do so, provided they came in from the north and were using Olbia as a base.
If Hitler does this, the British could find themselves caught up in a campaign that freezes up until American reinforcements (ground and air) arrive in 1943.

That the initiative could, however temporarily, pass to the Germans in the Med is a frightening thought.