Is the Body of Christ a royal priesthood?

No!  In II Peter 2:9, Peter uses phrases such as priesthood, and holy nation. Peter is speaking to Israelites, the godly remnant of Israel in his generation. He takes the wording right out of Exo.19 where God promised Israel she would be a kingdom of priests and a holy nation. We as the Body of Christ do not make up a nation. The Body of Christ is made up of people out of many nations. But Peter is writing to a (singular) nation, the nation of Israel (I Pet 1:1).