Sprint will not under any circumstance activate a competitor's phone. You could probably get away with it by activating it at the time of purchase at an Apple store, but if you bought this unlocked at say a Verizon store you're out of luck.
Also, something to note. Sprint SIM cards are a bit different. Sprint pairs their SIM cards with the device so that you stay locked to the Sprint network. Now your device is unlocked, so there may also be difficulty in making this work beyond Sprint's no activation policy.
Lastly, Sprint and Verizon do not share the same LTE bands. So, even if they agreed to activate your iPhone 5, managed to get it paired and activated, you would be unable to use Sprint's LTE.
The 5s is a bit different as it has more bands, but still, see the above about Sprint's policy.