It is possible to access many parts of most of these models, however I think it's not as easy as you might imagine to just dive in and improve them. Often, an individual needs a background in a specific scientific field in addition to some software engineering capabilities. Sometimes this means you have mathematicians and physicist pursuing higher degrees in atmospheric science, in other cases an ecologist gains a masters in CS. I think you might be underestimating the programming and numerical chops of people who are involved in making the models run, not everyone is a great programmer, but they do exist within the broad field.

The overarching climate models also typically aren't individual monolithic models, but tend involve various multidisciplinary pieces spanning much of the physical sciences. Also, get ready to work in Fortran. I spent a summer at Lawrence Livermore National Lab, and met some people there whose work was mostly the uncertainty quantification of outside climate models. My understanding at the time was that the lab specifically does not produce their own model, but rather, acts as an impartial testing ground for the major efforts from other global institutions.

All of that aside, they do have plenty of problems, and people from the outside do come in and make improvements, it's just not necessarily easy to break into. If you wanted something more accessible to start thinking about, you could do worse than the edGCM from Columbia[1].

[1] http://edgcm.columbia.edu/software2/edgcm-climate-modeling/