Learn Russian: Можно здесь постирать мои вещи?

With modal words like можно, using a perfective verb focuses on the result. The speaker doesn't just want to spend time doing the process of washing; they want the result of having clean clothes. If you used стирать, it would sound like you're asking if the general activity of washing is allowed here.

Doesn't вещи usually mean "things"?

Yes, вещь literally means "thing". However, in everyday Russian, вещи is the standard way to say "clothes" or "belongings" when talking about laundry, packing a suitcase, or moving.

What case is мои вещи here, and why didn't the ending change?

It is in the accusative case because it is the direct object of the verb постирать (it's what you are washing). Because вещи is an inanimate noun (not a living person or animal), its plural accusative form looks exactly the same as its nominative (dictionary) form.

Where is the word for "I" in this sentence?

The word можно means "it is possible" or "it is allowed" and doesn't require a subject. The sentence literally asks "Is it possible to wash my clothes here?" If you really wanted to emphasize that you are the one doing it, you would add the dative pronoun мне (to me) to make it Можно мне..., but here it's already clear from the context of мои (my).