How can I insert images into a Redactor rich text field in a front-end form?
It doesn't seems possible natively, you should write a custom upload script.

→ Example #1

→ Example #2

If it doesn't matter to add the images inside your rich text field, one more simple workaround could be to allow image upload to your assets using a separate upload field.

→ Uploading Files from Front-end Entry Forms