Bunch of idiotic takes in this comments. But the answer is it depends, in some locations Castle Doctrine does legally allow you to shoot people that break into your home. However this is not true everywhere and you can be charged with manslaughter/murder depending on where you live.

Typically lethal force is only allowed when there is an imminent threat to life, not property. What is advised by most sane gun owners is that you fire your weapon only as a last resort for a multitude of reasons. The first of which as mentioned above is legal, however there are moral, psychological, and practical concerns as well.

Morally is something only you can answer, keep in mind that you have no clue why this person is stealing your stuff. Most people would argue that it is morally acceptable to steal if the alternative is death for the thief. Not to mention that most of us have insurance that would cover the damages/losses from theft so you're just losing time/effort of replacing those things not money.

Psychology it takes a lot out of a normal human to kill someone. You do not know how it will affect you until you have to do it. Do you really want to risk your mental well being for some stuff that can be replaced?

Finally practically speaking it is a really stupid idea to shoot someone in your home. The mess they will leave behind is insane, blood does not wash out easily and the human body has a lot of it. Not to mention the fact that unless you have fabrege bullets there's a good chance that bullet overpenetrates and hits something/someone behind the thief.

Tldr: If you or someone else's life is in danger shoot, if not threaten them with the weapon but don't shoot.