Your color calibration is different/ you are using none. The previews were normalized to sRGB, the vector files contain the original color profile. Or they did something wrong on their end by not working calibrated in the first place...
Have you paid them? If not, you may sit on the monies until things are cleared up/amended.
As I see it, their AI file colours should match their previews, and they have the responsibility to create artwork that works for you; if there is something to fix, they have delivered an unfinished work.
It was finally resolved (one week later). To correctly view, I had to go to Edit -> Colors -> "Emulate Adobe Illustrator 6" and then all looks good in Illustrator. Is there anyway for me to resave this AI file in a manner that has this color information embedded in it? I guess now I can at least save high resolution PNGs, but it would also be great if I could open it in Inkscape and have the colors appear correct.
If inkspace has a color management feature probably just selecting the one you need. if not simply save it as an AI file or PDF or any format inkspace can open.