Learn Swedish: Annas mamma vet inte när festen börjar. - Anna's mother does not know when the party starts.

Because this is an indirect question, which acts as a subordinate clause. In a normal direct question, the verb comes right after the question word, as in När börjar festen? but in an indirect question, the subject festen must come before the verb börjar.

Why is there no apostrophe in Annas?

In Swedish, the possessive s is added directly to the end of the name or noun without an apostrophe. So Anna's mother is simply written as Annas mamma.

Could I use känner instead of vet here to mean know?

No, the verb veta and its present tense form vet are used when you know a fact or a piece of information, like when a party starts. You use känna and its present form känner when you know or are familiar with a person, a place, or a feeling.

Since this sentence contains a subordinate clause, shouldn't the word inte come before the verb?

The negation inte belongs to the main clause here, which is Annas mamma vet inte. The subordinate clause is only the indirect question när festen börjar. Because it is in the main clause, inte follows the normal rule and comes after the verb.