I know that some Christian organizations buy healthcare plans for their employees that don't include abortion coverage. But that's ultimately just a personal boycott. To the extent it aspires to make social change, it leaves the matter to market forces (and is a drop in the bucket).

A state-level fight over Medicaid contracts to pressure private insurers about what they do in other states would be another matter. I don't remember whether this has been tried to discourage abortion coverage, or what the outcome was.