Learn Italian: Mi faccio sempre stringere qualche camicia. - I always have a few shirts taken in.

Using the reflexive pronoun mi with the causative verb fare (the farsi + infinitive construction) shows that you are having the action done for yourself. So mi faccio stringere means I am having something taken in for me.

Doesn't stringere usually mean to squeeze or tighten?

Yes, it generally means to tighten or squeeze. In the context of tailoring and clothing, stringere is the standard verb used for taking a garment in (making it tighter).

The translation says a few shirts, so why is it singular (camicia) instead of plural (camicie)?

The word qualche (some/a few) is a special adjective in Italian that must always be followed by a singular noun, even when the meaning is plural. If you wanted to use a plural noun, you would need to use alcune camicie instead.

Why does sempre go between faccio and stringere?

Adverbs of frequency like sempre (always) normally go immediately after the conjugated verb. When there is a second verb in the infinitive, the adverb sits right between the conjugated verb (faccio) and the infinitive (stringere).