You don't need to be running Vortex after you deploy your mods. That's one of the great benefits. Just deploy the mods and close Vortex. Then launch SKSE64.
Note that if you rename skse64_loader.exe to SkyrimSELoader.exe (or whatever the exact names are) you can deploy your mods, close vortex, and then run skyrim exactly like you would run it through Steam, including using remote play and all that.
But it sounds like you didn't follow the instructions for installing SKSE and/or Vortex. Look in the vortex settings and make sure all the paths are where you expect them to be.
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