The searchBox widget of the standard InstantSearch.js library has a showSubmit option to toggle the submit button, but the React library does not.
The React InstantSearch widget does not have a building option to remove the searchBox submit button; but you can customize it. To completely remove the button, customize the widget using the connector.
Here's an example of how to do it: https://codesandbox.io/s/no-submit-no-reset-mkf9r