I know golems are powered by a bound earth elemental, which is basically their soul, but what about others? I would say mindless constructs don't, but ones that become sentient might "grow" a soul. I also think the awaken construct spell probably imbues them with a soul as well.
There are some that do, but most constructs do not. For example, the Occultist’s Soulbound Puppet is a construct with a soul. Awakened constructs night have souls, I’m u sure as the text is unclear on that one... Enlightened Constructs however have artificial souls.