I work in a marketing department and the word "Expedite" is used to represent the action of request a customer to provide an update on the status of a project e.g "can you please expedite the client?" when wat is meant is "can you please ask the client for a status update?".

Since expedite means "to hurry along" or "make happen faster", I can't help but think there is a better word to use in this context?
Something in this question does not make sense. Are you quite sure there is not some misunderstanding? "Can you please expedite the client?" does not make sense.

In business, asking someone to hurry up can be risky, especially if that someone is a paying customer.

"Please provide a status update at your earliest convenience." should provide enough incentive for most companies to get moving.