Learn Swedish: Kan du komma in?

Because it comes right after the helper verb kan (can). Just like in English, where we say 'can come' and not 'can comes', Swedish uses the basic infinitive form of the main verb after a modal verb like kan.

How does Swedish turn this into a question without a word like 'do'?

By putting the verb first! To make a yes/no question in Swedish, you just swap the subject and the verb. The statement Du kan komma in (You can come in) simply becomes the question Kan du komma in?.

I've also seen the word inne for 'in'. Why do we use in in this sentence?

Swedish makes a strict difference between movement and location. We use in when there is motion towards a destination, like coming inside. We use inne when someone is already resting or located inside. Because the person in this sentence is moving indoors, in is the right choice.