Is it possible to get the items on a specific artboard with script?

You can check for the bounds coordinates of your art and see if it fits within the coordinates of your artboard you want.

Or, you can activate your desired artboard and use the selectObjectsOnActiveArtboard() command to select all art on it and then cycle through that art to process the pieces you want. When using setActiveArtboardIndex as below, we use a 0-index based syntax, so for artboard #4, you have to put in a 3.

#target illustrator

function test(){

  var doc = app.activeDocument;

  doc.artboards.setActiveArtboardIndex(3);

  doc.selectObjectsOnActiveArtboard();

  var sel = doc.selection;

  alert("Your selection contains " + sel.length + " items.");

};

test();