The answer to this will be, for the most part, very similar to the answer to your question about taking care of a customer in a face-to-face situation. The steps that I can take will vary greatly depending on the exact nature of the customer’s problems.
When dealing with such a customer, the first thing to do will be to establish a tone that indicates that I care about their complaint. As much as possible, I need to have a tone of voice and use words that show that I am actively paying attention. I cannot use any body language here so I must be very careful about how I use words and tone of voice to convey this message.
To the extent possible, I need to resolve things quickly. I need to get detailed information from the customer. I then need to figure out what can be done. If this step is going to take a while, I need to tell the customer how long it will take. I should give them the option of holding or of having me call them back so I do not force them to stay on hold. I need to deal with their complaint quickly and I need to inform them of what I am doing or have done.
Without knowing the specifics of the complaint, these are general things that I can do to deal with a customer in this situation.