The NCR are a horde or invading body. Of course they have a "government run military" because they ARE a government run military, they are their military (I really wish I knew what this term was again...). Yet they have no police force because there is no separation of military and paramilitary units... and the closest we see are not government run but volunteer.
As well while they both have currency, they don't have established exchange rates associated with them.
They are kind of unsupported play money in all seriousness. Likely a way to delegate rations but their value is by all means open to interpretation. Which is why even within the NCR they often use the bottlecap system.
It is sort of a difference in scale.
In Fallout 2 you can clearly see that the NCR has an
established civilian police force and civilian government, with the military being kept entirely separate.
The NCR dollar is also the strongest currency in the region (gold-backed, given how it's literally golden coins) and is the game's default currency.
By the time of New Vegas, the NCR has begun to stagnate and the value of the NCR dollar has dropped considerably and they've switched from golden dollars to paper dollars, the value of which is lesser because for one, it's paper rather than gold coins and for second, it is not gold-backed anymore. Thus in the Mojave, Bottlecaps are used as they are a more stable currency.
The Mojave is a frontier for the NCR and thus most of the local judicial system has been left to the Rangers rather than the NCR police. Think the Westward Expansion in the united states (an apt analogy given New Vegas' distinct western tinge to the established post apocalyptic formula).
So, in Fallout New Vegas you don't see NCR police (save for the Military variety) because you're out on the frontier and a frontline for a big war. It's a region the NCR is clearly interested in and has somewhat begun to integrate into its territory, but that has as of yet not been entirely integrated and is also the front line of a big war.
Given how the war scared off most potential (and some existing) settlers, the rest of NCR assets (land, infrastructure, settlements) are under military rule due to circumstances. There are also local settlements that bargained with the NCR for protection, which is, again due to the war (and partially the distance from NCR's core lands) handled by the military and/or rangers.
Thus, most of the NCR people that you see are military personnel and most of the NCR assets in the region are protected by their army. and given how the war scared off most potential settlers, the rest of NCR settlements would be under military rule due to circumstances. Or they are local settlements that bargained with the NCR for protection (which again due to the war is handled by the military).
However I will gamble that Sharecropper farms wasn't built by the NCR and isn't what provides food for the entire New Vegas.
It doesn't, its produce is used mostly to feed the NCR military (some of the surplus is given to NCR refugees and squatters in Freeside. There's a quest related to that with the Kings). Most of New Vegas' food comes from trade with local farms and settlements, notably Westside.
However, the Sharecropper farms were built and are operated by the NCR. They offered money and protection to farmers willing to move to the Mojave to work the farms.