I have had the same problem for a long time.
Is there a solution in the meantime?
I have manually removed Windows write protection from the programme folder (c:\program files\Black Tree Gaming Ltd.\Vortex)
However, the automatic update still does not work. Only manual reinstallation helps.
I have no problems with Vortex itself or managed mods.
edit:
I was able to solve it for myself (as part of a new installation...)!
In the Vortex program directory there was a subfolder skrymse (with an older Unofficial Skyrim Special Edition patch).
I don't know how it got there because Vortex stores mods elsewhere. I had also uninstalled this version a long time ago and replaced it with a current one.
In any case, the automatic update worked perfectly after manually deleting this folder.