ArcGIS WebApp Builder (WAB) has a Grid Overlay widget which supports an MGRS grid. If you're not using WAB, but are using the 3.x API, you can download WAB, get the source code for the widget, and work it into your application.
If you're using 4.x, ESRI has no plans to migrate this widget to Experience Builder. In that case, you are left with the prospect of migrating the widget yourself.