Is it acceptable to omit 来 in 起来 as a direction complement?
起


  （用在动词后，表示动词的趋向）:
  
  hold high the flag; raise the flag;
  
  举起旗子
  
  take up arms;
  
  拿起武器


As you can see, the verb should be followed by 起 when there is an object at the end, for example 拿起武器. When there's no object, 起来 should be used, e.g. 拿起来. Basically, you can consider 拿起 as a transitive and 拿起来 as an intransitive. 

You can also use 把. It will be like 把...拿起来. e.g.  把武器拿起来.

So, 孩子们渐渐建立起对他的信任 can be paraphrased as 孩子们渐渐把对他的信任建立起来.