Believe it or not, the title—Al Capone Does My Shirts—isn't only paying homage to Piper's ill-fated laundry service. Though, of course, we don't realize this until the very end.
To recap, Piper's idea of a good time is to advertise a convict laundry service to the kids at school. It's basically the old-timey equivalent of having a celebrity do your laundry today… you know, if they were locked up and had no choice but to do so. Kids initially leap at the opportunity, but once the first batch is returned—and their shirts are still just regular old shirts—the wonder fades a bit. Like, who cares who washed it? It's just an ordinary clean shirt.
In the end, though, in an act of desperation, Moose writes to Al Capone (with a little help from Piper), asking him to use his connections to help Natalie get into school. And guess what? Just like that, Natalie does get into school, and as the story ends, Moose puts on a convict-cleaned shirt and finds a note in the sleeve that simply says: "Done." Looks like Al Capone does our main man's shirt—and got his message asking him to help Natalie.