If I was using an older OS X version that was purchased as a hard copy or be redemption code, would it work if I upgraded to Mavericks using App Store on any version OS X from Mountain Lion or earlier?

No, you can’t. If you’re running 10.6.8 or newer, you don’t need any intermediate OS to get Mavericks, but you do need to have access to an Apple ID previously used to download it, or to go to a physical Apple Store and ask for it.

Installing an intermediate OS(other than one needed to access the Mac App Store) won’t make any difference.

(155907)

The only older versions you can currently buy are Snow Leopard 10.6.3 and Lion 10.7.5. The former is sent on DVD. The latter by redemption code and download. Your computer model must be bootable from either. Mavericks can be downloaded for free if either is already installed unless your model is able to run a later version of OS X than Mavericks

Macs that can be upgraded to OS X Mavericks thru El Capitan

1. iMac (Mid 2007 or newer) - Model Identifier 7,1 or later

2. MacBook (Late 2008 Aluminum, or Early 2009 or newer) - Model Identifier 5,1 or later

3. MacBook Pro (Mid/Late 2007 or newer) - Model Identifier 3,1 or later

4. MacBook Air (Late 2008 or newer) - Model Identifier 2,1 or later

5. Mac mini (Early 2009 or newer) - Model Identifier 3,1 or later

6. Mac Pro (Early 2008 or newer) - Model Identifier 3,1 or later

7. Xserve (Early 2009) - Model Identifier 3,1 or later

To find the model identifier open System Profiler in the Utilities folder. It's displayed in the panel on the right.

Are my applications compatible? See App Compatibility Table - RoaringApps.

The only older versions you can currently buy are Snow Leopard 10.6.3 and Lion 10.7.5.

Mountain Lion is also available.

Mavericks can be downloaded for free if either is already installed unless your model is able to run a later version of OS X than Mavericks

Its availability is based on the logged-in Apple ID, not whether the Mac can run a higher OS than it.

(155908)

Yes, thank you. Forgot all about Mountain Lion!

The Apple ID would be a given inasmuch as the user may need to log into the App Store. Also, as given is that the hardware can be started with whatever OS X version one tries to use.