Is it legal to write a store manager a non-threatening letter to try and persuade him not to pursue shoplifting charges against your child? 

It is not illegal to write the store manager.  However, before you do so - especially if you're going to be making statements about your child - you should talk to an attorney.  Writing the store manager may weaken your kid's position if the matter is actually prosecuted.

In many of these cases, the store never actually sends anyone to court, which usually means that the case must be dismissed eventually because the state has no witness to the alleged shoplifting. 

Talk to a lawyer before writing a letter.

The decision whether or not to prosecute a case (i.e. whether or not to drop charges) rests with the prosecutor and not with the alleged victim.  What that means, from a practical standpoint, is that a case may be prosecuted even if the victim objects.  While the state's case would be stronger with the victim's cooperation, if there is other evidence to support the charge the case may still go forward.  So writing a letter will probably be of no help insofar as the criminal charge is concerned.  However, if your son is a first-time offender, he may be eligible for "diversion" (or your state's equivalent).  Basically, this allows for your son to be left with a clear criminal record upon successful completion of probation (or at least the ability to get his record cleared).  Bottom line, you will need to speak with an attorney as to all of this.  Frankly, they may be able to get the charge dismissed on a technicality; at the very least they will be instrumental in arranging diversion for you son.