By default, if you press Tab in the input field, the browser will switch focus to the next element. To override that behavior, you can listen to the keydown event and call e.preventDefault(), then add some code to insert the tab character at the cursor position. Below is the implementation. Note that because you are tampering with the input value, the undo feature doesn't work anymore:

<TextField
  multiline
  onKeyDown={(e) => {
    const { value } = e.target;

    if (e.key === 'Tab') {
      e.preventDefault();

      const cursorPosition = e.target.selectionStart;
      const cursorEndPosition = e.target.selectionEnd;
      const tab = '\t';

      e.target.value =
        value.substring(0, cursorPosition) +
        tab +
        value.substring(cursorEndPosition);

      // if you modify the value programmatically, the cursor is moved
      // to the end of the value, we need to reset it to the correct
      // position again
      e.target.selectionStart = cursorPosition + 1;
      e.target.selectionEnd = cursorPosition + 1;
    }
  }}
/>

Live Demo