Sadly, you cannot, from what I can tell.
The S-10 itself is a really good farm truck though, so I'd keep your eyes on the market for one. Or maybe the equivalent Ford one.
I'd like to have a Hilux, because the Tacoma isn't as good but that's just not an option here.
I have heard it claimed that they are not DOT compliant which prevents them from being sold to the public, so USPS crushes them all at the end of their life.
That said, once in a great while a non-USPS LLV shows up on GovDeals. Last one I saw was local to me a few years ago and it sold for over 10 grand.
For the very small bit of time (as you mentioned) - the Commercial Long Life Vehicle, or CLLV: https://www.flickr.com/photos/hugo90/33789231153
I assume you're talking about the LLV?
I had thought the post office bought a small number of 4x4 LLVs, but I don't see anything about it on wikipedia, so probably not. That said, maybe you could bodge something together from S-10 4x4 parts? (wikipedia says the rear axle is from the 4x4 s-10 blazer and the front axle is from the 2x4 blazer ... it might not be easy, but it's probably not impossible to grab a 4x4 front axle and stuff it in... )
I've read that the post office doesn't sell their old ones, but they weren't the only buyer and some of the other buyers do sell them.
Wikipedia does say the NGDV comes in all wheel drive. Although yeah, probably hard to buy a new one.