How to Update a Listbox on Another Form?

Declare a static variable (FormA SingletonFormA;) of the FormA (within FormA) which is accessible to both FormB && FormC.

Create a method e.g. CreateInstance(); within class FormA which has got something like:

public static FormA CreateInstance()
{
   if(SingletonFormA == null) // null check
   {
          SingletonFormA = new FormA(); // initialise it
   }
    
   return SingletonFormA;
}

Now create an object of FormA in each of classs FormB && FormC (ObjB, ObjC) and make a function call like:

// class FormB
ObjB = FormA.CreateInstance();

// class FormC
ObjC = FormA.CreateInstance();

Then, you can call something like:

ObjB.ListItem1.Add("testSingletonFromFormB", new ListItem("","")); 

ObjC.ListItem1.Add("testSingletonFromFormC", new ListItem("","")); 

The idea here is to use a shared variable amongst your forms.