You could look into it but I think the reason laundry chutes aren't made anymore has more to do with kids getting stuck in them and less to do with fire code (after all, stairways are chutes connecting floors).

Got stuck in a clothes a chute growing up, can confirm that those things are evil.

Of course there was another solution (I’m assuming you’re not still in the chute, by the way) - which is to make them so big that even a large adult male can’t get stuck.

But the main reason they’re gone now is that people put the laundry on the same floor as the bedrooms in modern homes.