There are competitions for subsets (PB, SAT, max-SAT) and for constraint programming, as you and other answers pointed out. You can find many competitions (DIMACS challenges, for example) with NP-hard problems that can be formulated as IP, too.
So, why are there no such competitions? My personal guess is that it comes down to:
- Implementation complexity. A good integer programming solver is HUGE and COMPLEX. SAT competitions and the like are interesting because many (small) teams can compete and a few tricks could get you quite far. There are only a few IP solvers, and all of them are many years of work.
- Too general. There are many many IP instances with different properties. It would be difficult to create a balanced benchmark set.
- Mature field. The solvers are mostly commercial, and the companies have no interest in organizing or taking part in such competitions.
There was a Pseudo-Boolean solver competition from 2005-2012, but (as far as I can tell) nothing since then. Integer Linear Programming is a subset of Pseudo-Boolean programming. See the 2012 competition page for results and links to other competition results.
It is not really a competition, but there is the well-known Mittelmann benchmark, which lists results for some of the best solvers (including commercial ones) for mixed-integer programming and related problem classes: