Jason,
The Local Layer widget is not a normal widget that you would click on a button to open or activate. It sounds like you did not review the install instructions that tell you to manually add the widget to the config.json. The odds.json is not a file that is included in the widget download but something that is manually added (if you want that ability). I have never seen the need to use the odds portion of this widget and it does not impact the functionality of the widget once properly installed.
Setting up the LocalLayer Widget
To add this widget to your ArcGIS Web AppBuilder environment, copy the widget folder to your \client\stemapp\widgets directory, and add a reference to the widget in the standard Default App configuration located at \client\stemapp\predefined-apps\default\config.json, under the widgetOnScreen section.
"widgetOnScreen": { "widgets": [ { "uri": "widgets/LocalLayer/Widget", "visible": false }, ... ] }
"visible":false will render the widget 'off' by default when building your applications. This gives you an option of whether or not you wish to override the default ArcGIS-Online-Map-based WAB functionality in favor of importing your own MapServices directly.
Please note that the "/LocalLayer/" portion of the uri path above must match the folder name of the widget on your local deployment, so if you're downloading the widget from github, you'll likely want to rename the folder from "LocalLayer-master" to simply "LocalLayer"