It's simple really - just take responsibility
I work in customer service, in a call centre actually so when people want to complain (and being a Supervisor) I've heard it all, you have to somehow appease them without that face to face reassurance.
I've always found the best customer service you can provide someone is to take ownership of their request. The one thing we all truly hate is when someone says I'll get some one else to help you, because I don't know. I know I always think, the next person is going to palm me off too!
I always like to tell the customer if I don't have the answer for them, but ask them to give me time to find out for them. That way, if anyone else ever needs to know the answer I'll have it for them too.
When doing this, because I am not face to face I always give my name and contact details. By giving the customer my contact details reassures them they are not going to hang up and never return their call, why would I, they can contact me remember.
I've always found this to be a great way to show the customer you are committed to helping them and offering them the service they request and deserve!