It seems to me that the vehicles from the other blocks have contributed to the wear and tear on the drain cover so it seems only right that they should contribute a share of the repairs.
I would contact their chairs and secretaries and tell them that you need to fix the drain covers and invite them to make their own suggestions about who to hire to do it, if only to make them feel connected and part of the process.
If there’s nothing in writing that makes them legally liable, they might get all bush lawyer about it and refuse to contribute. But most people are generally more reasonable than that and just asking them to share the cost, maybe on a pro rata basis, may get you the result you seek.
If not, you could tell them that since they don’t want to be involved they will not be given advanced warning of when the driveway will be closed so they can get their cars out. Let’s hope it doesn’t come to that.