I have work experience in local NGOs.
This refers to your experience of being in the local NGOs for your work. This more or less implies that you were an employee of those NGOs.
I have work experience with local NGOs.
This refers to your experience of doing work together with local NGOs, and it does not matter where you are working and for whom you are working.
I have experience working with local NGOs.
You didn't mention this one, but it is even broader and covers even the case that the local NGOs are subcontractors.