File: webkitRelativePath

<input type="file" id="file-picker" name="fileList" webkitdirectory multiple />
<output id="output"></output>

output {
  display: block;
  white-space: pre-wrap;
}

const output = document.getElementById("output");
const filePicker = document.getElementById("file-picker");

filePicker.addEventListener("change", (event) => {
  const files = event.target.files;

  for (const file of files) {
    output.textContent += `${file.webkitRelativePath}\n`;
  }
});