After some thought what a corporation offer, I come to the disappointing point that there is nearly none.
What if, the corporations can launch special missions for their corpmembers? Because not every corporation is very big/active, there would be different difficulties/sizes available. But only one can be choosen for the day.
An example of how a mission could look like for PvP:
Kill 50 enemy ships in PvP. Size 1
Kill 125 enemy ships in PvP. Size 2
Kill 300 enemy ships in PvP. Size 3
Every corpmember contribute with their kills to this goal. Like it was done in the faction PvP event - just for corporation now instead of factions.
I’m unsure about PvE. Maybe either score/wins could be the goal.
If the goal is fulfilled, every corpmember(or if wished only contributors) get rewarded. The rewards are depening on the difficulty/size choosen.
The limit to fulfill the goal is either a day-based system or a 24h-system.
OS is a bit tricky, because the way to do this is like creating an important transporter escort mission.
Only officer+ can launch such a mission and it will cost iridium to do so.
Maybe a corporation is able to launch a PvP and a PvE mission, but not two PvP/PvE mission at the same time(day/24h). So they have to choose wisely what difficulty/size of mission they are (think of being)able to fulfill and the reward for doing so.