Can .lyr files be used without shapefiles?

No you also need the data (which doesnt have to be a shapefile).

See: What is the difference between a shapefile and a layer file?

if a layer file is sent to a user on another machine without the data it was created from, it does not display on the map because it does not contain the source data. To get the data to display properly, the user must have the layer file and the shapefile it references

The real answer is more nuanced than @BERA make it.

In it's true that a .lyr doesn't contain any data but only the reference to the data source and the styling setting associated to it. That mean that you can't open/display a .lyr if you don't have ACCESS to the data source.

BUT a .lyr can reference local OR online data, if it reference a local data source you need to ask for the source data but if your .lyr reference an online resource you only need an internet connection to open it (and of course ArcMap...).

For example if you go to : this page and click on "ArcMap" in the "View In" section, you will download a .lyr that will open in ArcMap...