Google-sheets – Pull data from one sheet (has a locked cell) to another sheet (has a locked cell) every day automatically

You could use an onEdit trigger for the script. That will update whenever changes are made. For example:

/**
 * A function named onEdit will be called whenever
 * a change is made to the spreadsheet.
 * 
 * @param  {object} e The edit event (not used in this case)
 */
function onEdit(e){
  var copyFromRange = 'Sheet1!A13:A'; // no row for second cell reference
  var copyToRangeStart = 'Sheet2!A3';
  copyValuesOnly(copyFromRange, copyToRangeStart);
}

/**
 * This function will copy the values from a given range to
 * a second range, which starts from the given cell reference
 * 
 * @param  {string} copyFromRange    Range reference eg: 
 * @param  {string} copyToRangeStart Cell reference eg:
 */
function copyValuesOnly(copyFromRange, copyToRangeStart) {
  var ss = SpreadsheetApp.getActiveSpreadsheet();
  var source = ss.getRange(copyFromRange);
  source.copyTo(ss.getRange(copyToRangeStart), {contentsOnly: true});
}

See https://developers.google.com/apps-script/understanding_triggers for more details on triggering scripts automatically.

I can't comment yet, so will ask here: Do you need to copy to a second spreadsheet or just a second sheet within the same spreadsheet? The second case is answered above, but I can adjust if you need to copy to a separate spreadsheet?

edit: added code to fully answer the question. Thanks for the guidance Jacob.

Get the sheet that was edited. Change code to:

var sheet, ss;

ss = SpreadsheetApp.getActiveSpreadsheet();
sheet = ss.getActiveSheet();//Get the active sheet

Now the sheet variable is the active sheet that was just edited. So anything done with sheet will affect that sheet.