To "have a shot at" a prize or reward means that your chances of winning are not bad. It means that you think there's a chance that you might win.
You can use this phrase followed by a noun:
Do you think I have a shot at the job?
Or you can follow it with "___ing":
Do you think I have a shot at getting the job?
There's also a similar phrase that's used when talking about men trying to attract women. When you think a man might be able to attract a woman, you say that he "has a shot with" her.