Approximately 5% of the Italian population has received at least one dose of the vaccine, as of today, after 2 months from the start of the vaccination program.
At this pace the government forecasts that the goal of vaccinating at least 70% of the Italian population would be reached in March 2023. Two real time sources from two reliable newsgroups are linked below.
Currently Italian localities are classified in colored tiers depending on the gravity of the pandemic data and viral spread: white, yellow, orange, red (from minimal to severe). The classifications changes weekly. Restrictions in a red zone means that nobody can be outside at any time except for specific necessary reasons. In an orange zone, you can be outside but only within you city limits.
In any case, as of today, you would not be authorized to enter the country except for proven work, health, or emergency needs (and even in that case, good luck finding flights, since airlines continually cancel scheduled flights).