Which trade(s) should I hire for structural repairs?
Talk to your city to see what they are going to require.  The inspector should come out and assess the home and see what work needs to be done.   There is a good chance that your inspector may say nothing.  I am not sure where on the joist the holes are drilled but really we are talking about something that is completely insignificant 99% of the time.  Also when it is significant it isn't like your joists will collapse.  They will simply start bowing, you will get cracks in drywall or dips in flooring.  I have never heard of a joist snapping due to a 1/2" electrical hole (I have also never heard of one bowing because of this).  

So the inspector will probably do one of the following:


Says don't worry about it.
Says I have no idea what that means, hire an engineer (I really feel for you if your inspector is like this).
Suggests a type of bracing.  There are lots of braces and straps meant for situations like this.  They take about 2 mins per brace to install.  I have used braces like this for plumbing, albeit never for electrical.  


Who you hire?  Depends on the answer you get.  If #3 then any general contractor will work,  if #2 then try to get a general contractor with experience doing this or one that has a relationship with an engineer to make it cheaper.  For example I could give an engineer I use your house specs and issues and he would OK a brace in about 5 mins because he knows potential deflection of joist due to hole and the amount the brace can handle - him knowing these braces are well over what is needed he wouldn't need much to sign off other than seeing pictures of the joists.  But that is him knowing I would do the work.  If I just contacted any structural engineer in the yellow pages it might cost me 500-1000, since I use someone all the time he might charge $50 or make me pay for next dinner.