I guess you could try to go for the tortious interference angle:
I have a contract with the customer which involves interacting with them by email. Customer uses EmailCompany. EmailCompany unfairly blocks me from emailing customer. EmailCompany is interfering with the contract. Both myself and the customer have a reasonable expectation that we should be able to carry out our contract by emailing back and forth, so EmailCompany is obstructing the contract.
Any of this sort of "advocacy" only works if the customer actually cares about receiving your emails though.
In my experience, the customer will just switch email providers when I tell them we're blocked. I had an email provider just the other day who wasn't accepting my emails. I tried to get them to unblock us, but to no avail. I told my customer and he went and registered a new email elsewhere.
I have a contract with the customer which involves interacting with them by email. Customer uses EmailCompany. EmailCompany unfairly blocks me from emailing customer. EmailCompany is interfering with the contract. Both myself and the customer have a reasonable expectation that we should be able to carry out our contract by emailing back and forth, so EmailCompany is obstructing the contract.
Any of this sort of "advocacy" only works if the customer actually cares about receiving your emails though.
In my experience, the customer will just switch email providers when I tell them we're blocked. I had an email provider just the other day who wasn't accepting my emails. I tried to get them to unblock us, but to no avail. I told my customer and he went and registered a new email elsewhere.