Yes, communication about scheduling, appointments, and suchlike is often ambiguous in this way. See, for example Schedule on Tuesday, and note that the two answers interpret 'schedule on Tuesday' differently.
In the OP's case, within 24 hours can be taken to qualify agree or appointment. Under the former reading, what is important is whether the agreement to the request for an appointment is provided within 24 hours of the request, and nothing has been implied about when the time that the appointment would be for (it could be for a time far in the future, for all we know). Under the latter reading, what is important is whether the appointment itself is for a time that is within 24 hours of the request.
It can be argued that the fact that within 24 hours is closer to appointment than to agree makes the latter reading more reasonable, i.e. that if the former reading had been intended, the formulation would have been 'agree, withing 24 hours, to an appointment'. That argument has some force, but is unlikely to be decisive, particularly when one is analysing something that has been said in a quick flow of casual communication.
So, the sentence is ambiguous when considered on its own; it can be disambiguated only if one has some further knowledge of whatever it is about and the surrounding circumstances.