The catch:
NASA should award prize money to the top 2 finishers with privately developed orbital craft. Same for a water seeking expedition to the Lunar poles.
Would this work? Sure.
And how many NASA personnel would this require? A few accountants? How many NASA facilities would this keep running? A suite of offices?
See the problem? As with many government programs, it is easy to forget that their main function is not the one that is advertised. NASA's primary job is not getting people and science equipment into space. It is to keep NASA large and well-funded.