Does every NP problem have a poly-sized ILP formulation?
This answer is mostly a recap of the comments on the question above.

If a problem is NP-complete, it can indeed be reduced to ILP, by using Karp's reductions (― Joe, andy). Claims of "polynomial sized formulations" from one problem to another, are likely meant as more direct formulations, as opposed to multiple reductions through SAT (― Kaveh).