The map editor is designed to run only in the admin. You could do this as a workaround:
1. On the frontend let the users just enter their address
2. Create or update a post on the backend for the business. In that post, populate a custom field ‘address’.
3. Configure the MapPress settings to generate a map from field ‘address’
This way a map will be generated in the business post when it is created or updated on the backend – not as nice as seeing it immediately, but it may work for you.