How can I replace Dojo's DomConstruct.place() with vanilla Javascript in the following snippet?
DomConstruct.place(this._rect, this._widget._xComponent, "after");
I'd recommend learning about DOM manipulation, specifically moving and placing elements, here.
That said, in order to replace Dojo's DomConstruct.place() with vanilla Javascript you would need to do something like the following:
this._widget._xComponent.appendChild(this._rect);
You may need to reference the nodes differently or select the nodes by id using document.getElementById("myId"), but appendChild(node) is what you are looking for.