Welcome to the forum!
Yes, you can use gradient colors, and it's still a vector file. And Inkscape can do it! Although the image you showed us is a raster format, not vector. And just importing it into Inkscape won't make it vector. It's possible you just showed us a raster version, and you also have a vector version. But if raster is all you have, then it will have to be converted to vector.
There are 2 ways to do that. You can auto trace it, using Path menu > Trace Bitmap, or you can manually trace it with the Pen/Bezier tool. The gradient will not auto trace - at least not keeping the gradient - it will come out as a rainbow of individual colors. So if the version you mentioned as being only outlines is not already vector paths, and you want to auto-trace it, I would suggest to color it some solid color. Then after it's traced, you can remove the solid color and replace it with the gradient. Or you could just trace with the Pen tool.
If you already have vector paths, you just need to add the gradient. Then File menu > Save As > EPS. I see you're using a Mac, so you should be able to do that with no problem. But Inkscape cannot save as AI.
Edit
If the image you showed us is the full size, you'll want to find a larger version, to get the best results from auto-tracing. Images that small tend not to trace very well.
But it's a simple enough image that tracing with the Pen tool shouldn't take very long.
