I am using LibGDX to make a very simple game. I am going to hire someone to create a splash screen logo since I'm artistically challenged. I want to order a vector graphic so I can scale it later if needed. What would be a decent size to use as a baseline (in pixels) that would scale relatively well on desktop/android/ios/etc?
Vector graphics are completely resolution-independent. Which means you can always scale them up as much as you like. There are no resolutions which scale especially well.
Vector programs like Inkscape use the px unit, but that is only the same as the output size if you output as 90 dpi. In any other case, the image will stay completely sharp, but the pixel units used inside the vector graphics program won't correspond to the pixel size outputted.
What you should care about, though, is the aspect ratio. Modern phones mostly 16:9 (landscape of portrait), while some Windows Phones have 15:9 (actually, the majority of Windows Phones). Tablets can also have other aspect ratio, mostly a bit squarer, like 16:10 or 2:3 (the Surface Pro 3). Therefore, I'd recommend you to tell the artist to fit everything inside a 16:9 aspect ratio, but put something above and below that 16:9 box so you don't have to display black bars on other aspect ratios.