Can I use
?

Filesystem & FileWriter API

- UNOFF

Method of reading and writing files to a sandboxed file system.

IE

  1. 5.5 - 10: Not supported
  2. 11: Not supported

Edge

  1. 12 - 18: Not supported
  2. 79 - 85: Supported
  3. 86: Supported

Firefox

  1. 2 - 80: Not supported
  2. 81: Not supported
  3. 82 - 83: Not supported

Chrome

  1. 4 - 7: Not supported
  2. 8 - 12: Partial support
  3. 13 - 85: Supported
  4. 86: Supported
  5. 87 - 89: Supported

Safari

  1. 3.1 - 13.1: Not supported
  2. 14: Not supported
  3. TP: Not supported

Opera

  1. 9 - 12.1: Not supported
  2. 15 - 70: Supported
  3. 71: Supported

iOS Safari

  1. 3.2 - 13.7: Not supported
  2. 14: Not supported

Opera Mini

  1. all: Not supported

Android Browser

  1. 2.1 - 4.4.4: Not supported
  2. 81: Not supported

Blackberry Browser

  1. 7: Not supported
  2. 10: Supported

Opera Mobile

  1. 10 - 12.1: Not supported
  2. 59: Supported

Chrome for Android

  1. 85: Supported

Firefox for Android

  1. 79: Not supported

IE Mobile

  1. 10: Not supported
  2. 11: Not supported

UC Browser for Android

  1. 12.12: Not supported

Samsung Internet

  1. 4: Not supported
  2. 5 - 11.2: Supported
  3. 12.0: Supported

QQ Browser

  1. 10.4: Not supported

Baidu Browser

  1. 7.12: Not supported

KaiOS Browser

  1. 2.5: Not supported

The File API: Directories and System specification is no longer being maintained and support may be dropped in future versions.

Resources:
Firefox tracking bug
HTML5 Rocks tutorial
WebPlatform Docs