I have the exact same problem since upgrading Vortex from 0.13.0 to 0.13.5 yesterday.
Suddenly, when I try to launch SKSE (which is set to primary, so either from it's box, or the main Skyrim SE box), I get the first requester box from Vortex asking if it's ok to "Run as administrator", then I get the system requester asking "Do you want to allow this app to make changes to your device?", and after I say "yes", the pointer arrow just flickers a couple of times and that is it. No cmd window ever briefly opens and closes, like normal.
When I double click the skse_launcher.exe icon in my Skyrim SE directory, I get the system requester "Do you want to allow his app to make changes to your device?", and after I say yes, it briefly opens and then closes a cmd window, then a while later Skyrim SE loads up.
I did notice there is an extra backslash in the "Start In" box on the SKSE launcher in the Vortex dashboard, so the path reads:
"C:\Games\Steam\steamapps\common\Skyrim Special Edition\\"
But I don't know for sure if that was that way before I not (I think it wasn't). I tried deleting the extra backspace and saving, but it still doesn't launch Skyrim SE.
The Target box is currently:
"C:\Games\Steam\steamapps\common\Skyrim Special Edition\skse64_loader.exe"
Which used to work...