I have been playing a lot of Salmon run this past weekend and I am not sure how I get the prizes. Is there a special place where you have to pick them up?
Since I own the German version I don't know how it is called in English, but when you stand in front of the building where the Salmon Run is played, you will find a counter to the left of it. There you can pick up your prizes.
At the end of every round of Salmon Run, after it shows how many Golden Eggs you got, in the bottom left corner there will be a bonus meter. You fill this up by going over your quota. When you fill the bonus meter up, go to the left of the place where Salmon Run is played. There will be a counter. You get them there.
As you complete games, you get points. They count towards your rank and bonuses (prizes/items) you can unlock.
They go to your inventory immediately, I'm pretty sure.