Does this description of verifiable random functions seem accurate?
Sure VRFs can be used for sortition. One can say that a party is selected if the output of his VRF satisifies some property. 

The VRF outputs two things - a random output and a proof that the output is computed correctly. It is not always the case that if the output of a VRF is f(x) the corresponding proof is x. For example, the VRF could simply output a pseudo random function $PRF_{sk}(x)$ along with a zero knowledge proof that the PRF was computed correctly. 

"but impossible for them to figure out how the function actually operates in general" : 
This would not be accurate, everyone knows how the VRF operates, else how would one verify that the output is actually correct. In cryptography the working of a protocol is never hidden (Kerckhoff's principle)

You can find more information on VRFs here. This is a working IETF draft to standardize the construction of VRFs. They have some interesting constructions there. 

Welcome to the community!