When using the non-developer edition of Web AppBuilder, is there a way yet to reorder the basemaps in the basemap gallery widget? Currently if you add a new basemap, it gets added to the end of the line, but there does not appear to be a way to reorder them, short of deleting them all, and adding them back in the order you want them displayed.
Another way is to use the custom templates group. if you save all your basemaps individually as maps saved to the custom templates group (you have to set your default basemap gallery to look to that group), you can use a "1-", "2-", "3-" prefix infront of the basemap name. When you add the basemap widget, it will honor that numbering as it sorts the basemap tiles. You simply have to click on each one and remove the number in WAB. If you want to reorder the basemap tiles, simply go back to your custom templates group, and change the number pre-fix, then go to WAB and remove the basemap widget, and re-add it, which will prompt it to reorder the basemap choices.
So that is my duct tape and bubble gum method, which will take just as long up front as adding them manually into WAB, but will be much quicker in the long run especially if you have multiple applications you are building.