Learn Turkish: Müdürden izin istedim. - I asked the manager for permission.

In English, you ask someone for something. In Turkish, when you request something, you literally "want it from someone." That is why you add the "from" ending (the ablative case -den/-dan) to the person you are asking.

I learned that "to ask" is sormak. Why does this sentence use istedim (I wanted)?

You use sormak when you are asking a question to get information. When you are asking for an object, a favor, or permission, you must use istemek (to want / to request).

Why doesn't the word izin (permission) have the accusative case ending here?

Because you are asking for permission in a general sense, not one specific, previously mentioned permission. When a direct object is general like this, it stays in its plain dictionary form and usually sits right next to the verb.

How is the verb istedim built?

It comes from the verb istemek (to want). You take the root iste-, add the past tense marker -di, and finish with the "I" person ending -m.