Interaction between Vilis, Broker of Blood and Phyrexian Delver
They happen in the order they are written, so you will get to draw cards.


  
  Resolving Spells and Abilities
  
  
    608.2. If the object that’s resolving is an instant spell, a sorcery spell, or an ability, its resolution may involve several steps. 
    
    
      608.2c The controller of the spell or ability follows its instructions in the order written.
    
  
  


and


  603.2. Whenever a game event or game state matches a triggered ability’s trigger event, that ability automatically triggers. The ability doesn’t do anything at this point.


So at the time that you are losing life, Vilis, Broker of Blood is already on the battlefield; it's triggered ability triggers just as if it had been there even before Phyrexian Delver was resolving.