Replace plywood behind outdoor meter can
See @Tyson's comment - Let's start with your comment:


  On the one hand this seems like an easy DIY project.


Hate to be the bearer of bad news but its not. Note that I and others can see all kinds of things that also need to be repaired electrically. Also in order to pull the meter base and enclosed breaker of the board you want to replace you will need to call your utility company for a disconnect/reconnect. This means that the project will have to be permitted and inspected in most areas of the county (USA). 

A good electrical contractor can handle the work including the material to hang it on to fit your location. Beware there may be other damage to structural timber if moisture has penetrated the wall and rotted them out.


  How would you go about getting this done with reasonable expense?


What do you consider a reasonable expense? In my area something like this would run around $1500 to $2000 for repair and that would not include repair of any unforeseen damage.

Hope this helps and good luck