Kit's aunt and uncle were not waiting for Kit at the docks when the ship, The Dolphin, arrived at dawn because Kit never actually wrote her aunt and uncle to say she was coming. Kit had lost both her parents years before and had been being raised by her grandfather in Barbados up until he passed away too. Orphaned and now all alone in the world, Kit decided to run to the only family she had left. In Chapter 2, we see her explain her situation to Captain Eaten in her speech:
My uncle and aunt can hardly be blamed for not meeting me. You see--well, to be honest, they do not even know I am coming.
We learn in Chapter three that Kit absolutely had to leave her home in Barbados as her grandfather lost their estate and all their fortune due to his prolonged illness. We further learn that Kit felt both desperate and afraid. She was desperate to find a new home, and she felt terrified of the prospect of being rejected from her aunt and uncle's home. Hence, because she was so afraid and so desperate, she did not dare write to her aunt to tell her of her need to come and instead came without warning.