What case you need depends on what place/function the pronoun fills in the clause.
In this case, it's an object. The relevant verb is schildern, and its objects are:
jemandem (=Dativ) etwas (=Akkusativ) schildern
to describe something to someone
The case of the pronoun you need depends on which of these objects your pronoun is going to represent. In this case, it's the dative object because it's the people at the Ärztekammer to whom the problem is to be described.
So that's why it is Dativ pronoun "denen".
(A bit special in this sentence is the fact that the plural form "denen" is used although "Ärztekammer" is singular. But I think that's the same in English - the medical association consists of people whom you hope to talk to.)
If in doubt about the case, it might or might not be easier to first replace the pronoun with the noun it stands for and look at which case you're using:
Vielleicht kannst Du bei den der Ärztekammer anrufen und den Leuten dort das Problem schildern?
But I guess it depends on your familiarity with the language whether that does in fact help you or not.
Other examples for cases in relative/demonstrative pronouns:
Der Mann, den (=Akkusativ) ich in der Bank gesehen habe, trug einen schwarzen Hut. (because: Ich habe den Mann gesehen. "Den Mann" is the accusative object.)
The man whom I saw in the bank was wearing a black hat.
Am Telefon habe ich mit Frau Müller gesprochen und nach deren (=Genitiv) Vorschlägen gefragt.
I talked to Mrs. Müller on the phone and asked for her (=genitive) suggestions.
(BTW: when you have understood this topic, you will never have any problems to decide beween "who" and "whom" in English either. Or, if you understand the difference between who and whom, this topic should be easier to grasp in German, too.)