Is there any way to install a non-Windows Store program I downloaded, on a Windows 10 S PC?
Windows 10 S is locked down and only approved apps from the store can be installed.
You have to convert to PRO if you want the ability to install apps that are not from the Windows Store. There is no other supported method for getting around this.
Info on that process: https://www.bing.com/search?q=upgrade+windows+10s+to+pro
As at the time of writing, it seems that you can switch some Surface Laptops from Windows 10S to Windows 10 Pro for free, but not others.
For instance, on my Surface Laptop (i7/16GB RAM) if I type “Switch out of S mode” in the Windows Store search mode, I get an option to do just that, for free:
On my daughter’s Surface Laptop (i5/4GB RAM) if I type that exact same search I don't get that option.
And I can see an option for my i7 under Settings to "Change the Product Key or upgrade your version of Windows". Whereas I get no such option for the i5
On my daughter's PC if I try and download some software from the web, I get some message saying I can’t do that, but with a link saying something like “Still want to install software from outside the store? Click here”. But click on that link, and I get taken to the Windows Store, which then displays a message along the lines of “Couldn’t load page”. I logged a bug on that site, and saw other folk were in the exact same boat.
I did see one thread somewhere saying that the switch is only possible on the more expensive Surface Laptop SKUs (but not going into specifics). This looks like the issue for me, and perhaps for you too.
On Microsoft’s site, they say:
You can switch out of S mode easily at any time through the Microsoft Store within Windows. There is no charge for switching the mode. The switch is one-way; if you make the switch, you won’t be able to go back to S mode. See below for important additional limitations when switching out of S mode on devices that use the Qualcomm Snapdragon processor. To learn more or switch out of S mode, go to the Microsoft Store within Windows and search for “switch out of S mode”.
…but obviously Microsoft don’t actually own any of their own i5 computers.