This phrase is used in the book shortly before the altercation between Johnny and Pony and the Socs. This is the fight in which the Socs start to beat up on the two greasers for having been with the Soc girls. It ends up with Johnny stabbing Bob to death.
In this context, the phrase “reeling pickled” simply means “drunk.” There are a lot of ways to say that someone is drunk in the English language. One of them is to say the person is “reeling.” This is because people who are very drunk often do not have enough control of their bodies to walk straight. We sometimes say they are “pickled,” presumably because we are saying they have been soaked in liquid like a pickle is.
When Ponyboy sees the Socs, they are staggering. They are not walking straight. Therefore, he thinks they are drunk. Later on the same page (this is p. 54 in my copy of the book), Ponyboy tells us that
They smelled so heavily of whiskey … that I almost choked.
This helps us to understand that “reeling pickled” means “drunk.”