I see where you're coming from, it does appear that Hamlet is striking out on his own in his revenge plot to avenge his father's death. Everyone thinks he's crazy, including his own mother. His friends have turned against him and sided with Claudius and they are trying to help Claudius bring Hamlet down. Hamlet must feel alienated in this play because no one believes him and worse yet, while he's out trying to get Claudius, everyone is out to get him.
I think Daveb has some good points, but everyone is always dealing with their own struggles- that's nothing new or old. I agree that almost every character was dealing with some major vengeance, but since the play is primarily about Hamlet's own plot, I think it's safe to say that he is primarily the one seen alone trying to seek revenge.
Taking vengeance happens in a variety of ways in Hamlet, though it happens most clearly in Hamlet's actions.
You could, for instance, easily include Fortinbras as someone taking vengeance for the death of his father.
Does Ophelia take vengeance for the acts committed upon her by killing herself? Perhaps, assuming you believe that she killed herself.
Does Claudius take vengeance upon his brother for some reason when he kills him to become king? Perhaps, though he certainly coveted Gertrude.
Does Laertes at least attempt to take vengeance upon Hamlet for his sister and father? I think so, though he does see the error in his ways.
Does the Ghost, Hamlet's father, take vengeance by letting Hamlet in on the secret? I think a case could be made for that.
I think Fortinbras and Laertes are your best contradictions to the opening statement.
Hamlet is alone to an extent as he is the only avenging character who contemplates his following actions "to be or not to be" soliloquy but i think that it is still extremely important and clever of shakespeare to have these other contrasting characters like laertes and fortinbras so the audience is able to understand how perfect Hamlet is to play the role of the avenger and is the only one who truly fulfills this as laertes seeks forgiveness from Hamlet and fortinbras only regains his fathers land by becoming king, which Hamlet gave to him in his last moments