First off, don't bother with what /u/Little0mid is saying. You can definitely make sample based jazz. Herbie Hancock, Miles Davis, Flying Lotus, Madlib, Meshell Ndgeocello and others have done it very successfully.
The rest of his comment is just trying to act superior because you didn't know specific terminology. Technically he's right but don't worry, any jazz musician would have been able to understand you. Yes, he's right that type of feel is called swing, not shuffle which usually applies more to blues. Also what you referred to as smooth jazz is actually more like cool jazz. Smooth jazz would be something like this.
As to how you can make this album, I'm a jazz musician and a producer so I think I can help you out. For VSTs you don't want anything too crazy. If you're making jazz everything should be pretty natural. That's not to say you can't use synths but they have to have the right texture. Here's some examples of great use of synths in jazz:
You can also use bass synths to mimic a bass but only if it's the right style. For example it'd be possible to recreate a bass like this, but probably not one like this or this. There's lots of quality pianos out there and it's not hard to make a good electric piano in Massive or something similar. Look up some tutorials on that, they'll explain it better than I can. Drums you should probably chop out of jazz records (remember the names Art Blakey and Max Roach next time you're at the record store). Saxophones, trumpets, trombones, etc. are impossible to recreate faithfully with synths. You'll either need to sample those or find somebody who can play them for you. You can find great jazz records really cheap at any record store.
Lastly, most importantly, you have to listen. You can't start a project like this without really being familiar with the music. Learn the history of the music and how it came to be what it is. I believe this sub has an essentials list that can help you get started. Check out rateyourmusic.com too, there's lots of lists there that could help you get started.