It can be done in vanilla PzC, but it needs a few changes done in campaign.pzdat.(and it needs to be stressed that backups should be done to any file BEFORE you try doing anything else with it)
Multiple nation campaigns are now supported - just list all nations separated with commas in the campaign.pzdat file.
From the official changelog of the 1.10 update:
viewtopic.php?f=121&t=36277
The id number for Italy is 5 if I remember correctly and it needs to be added to each of the campaigns that are listed in campaign.pzdat, separated with a comma from the existing 0 id which is the one of Germany.
However, multinational campaigns do not work the way they did in PG2 iirc. Afaik in PG2 you could buy units that were not of the main/core faction if you encountered such allied units in a scenario and there were flags/recruiting centers of that allied faction on the map. Then you could take those units with you in the next scenario and place them on the map. Their faction flag I think would only appear in the purchase screen if you used those units to take over some recruitment center on the map.
In vanilla PzC if you enable multinational campaigns by editing the campaign.pzdat alone you`ll get the following results:
1. you can buy and use non-german units provided they are part of a scenario and you have cities with their flags to place them.
2. you can take those units with you to the next scenario and use them on the map.
3. you will also get an announcement each time units for any faction allowed in the campaign become available. This means for example that if you have enabled Italy in the 1939 campaign, when you get to Norway you`ll be shown the newly available Italian units, even though Italy has not entered the war yet.
4. the "allied" units that you`ve purchased can be reinforced(including with elites), but you can`t buy more of them unless you again arrive at a scenario that features their faction. Even if you use them to occupy an enemy city their flag won`t appear in the purchase screen because:
5. the flag placed in the city occupied by them does not actually seem to be recognized as a flag for the side(Axis for example) on which you are fighting. It does not allow units to be bought there and if you move a german unit on that city/airport the flag will become german as if you had just conquered it.
To address this you actually need to edit each scenario of the campaign and add the faction you want(Italy in your case) as a combatant on your side. That has the effect that Italy(in our example) will appear in the purchase screen on each scenario edited this way, whether you actually have italian units in your army or not.