Unless this is supported in Project Server, of which I have no experience, you cannot do this in MS-Project. It simply does not include the concept of resource dependencies within a task.
In any case, despite the fact you say you do not want to split the task, the fact that it has two parts fulfilled by two separate resources with a strict linear order, means that the overall task does consist of two subtasks. If that is the reality then modelling it any other way in MS-Project does not make sense.
I don't know whether it is any more palatable to you, but if I were doing this I would create an umbrella task and indent one level for the two sub-tasks which you can then order using a Finish-to-Start constraint. Then just use the umbrella task as if it were one task, i.e. it have its own constraints and scheduling (to a certain extent), it just happens to be made up two separate tasks internally.