I think the story could really be improved if you read this back to yourself. A lot of the sentence structure is oddly put and the action. For example:
I heard a zombie right next to me. I jumped to the left, and turned to shoot. It was about to bite me as I shot it through its left eye before it collapsed in front of me.
This almost sounds the zombie was going to collapse in front of you, but before that happened it almost bit you. Also, you took a moment to shoot it in the left eye while this was going on.
I heard a zombie right next to me. As it tried to bite me I jumped to the (side), turned, and shot it through the left eye (eye socket would sound more gruesome). It collapsed in front of me.
That sorta makes the sequence of actions more clear. Lot of other examples, but I think you could easily reword them if you went back and read it aloud as they would sound awkward to the ear.
The other thing narrative wise that I noticed is the over use of the word "I". Everything is "I did this" then "I did that.
I ran to the house and immediately went inside. I locked the doors and windows to stall the zombies as I pillaged for food and car keys. I noticed they left a pistol on the couch with a note. I quickly read the note,
Most of the time you can leave out the "I" part and it will still make sense. Also it brings a sense of immersion for the reader. Sometimes while putting the story to paper you will develop the habit saying "I" for everything because it is basically you describing to
yourself how you would do things as the protagonist. One trick is to write what is going on in the protagonist's head during this time he is doing so much stuff.
Quick! I need to get to the house! I locked all the doors and windows in order to stall for time and pillage (gather... it is his house right?) food and supplies. Where are the keys...!? There was only a pistol left on the table and a note. It read:
One last thing I wanted to mention was with the story itself. This chapter was an intro to our protagonist story. The intro has to give us a reason to want to read on besides, "Well, there are zombies and danger! Isn't that something!?" In this case it seems the reason to read on is because the protagonist is going to try to meet up with his family. If that is the case, then that needs to be fleshed (pun) out more. He saw them drive away. Did he care they left him? Did anyone look back at him? Were they scared or were they a professional zombie killing family? And how does the hero feel? Does he even like his family? He just sorta stared at them as they drove off. Is he feeling desperate or is he calm?
You don't have to go on long explanations to answer any of these questions but you should give the reader a hint. If he is desperate, have him run after the car when he sees his family driving away. If he is calm, show him being non-chalant when they leave without him. Just a tiny bit of insight into his thoughts will help the reader feel closer to the protagonist which in turn creates immersion.
And that is the key to a good story - especially horror.