The sentence above is perfectly grammatically accurate. It is a request for a colleague to do something for the patient. The difficulty is, because it is phrased as a request with a question mark at the end, the implication is that your colleague could say no.
Often when making requests of colleagues on behalf of patients we do not see them as optional. We want or even need them to happen. Rather than phrasing them as request questions, it is better to write them as request statements.
For the example above, this would change the question to:
Please arrange a follow-up appointment for him.
The request statement is still polite but now it is clear to the reader that you expect them to complete the action.
Here are a couple more examples:
Would you be able to contact his next of kin? → Please contact his next of kin.
Can you please provide a second opinion? → It would be appreciated if you could provide a second opinion.