This comes down to who is ruling House Arryn.
Originally the Lord of house Arryn was Jon Arryn, Robert Baratheon's hand.
Jon Arryn mysteriously dies, which is what initially brings Robert to the North, to recruit Ned Stark to be Robert's new hand.
When Jon Arryn mysteriously dies it leaves Lysa left to rule over the Eyrie. Lysa isn't know for her mental stability and becomes even more paranoid when her husband dies; so she locks herself and her child, Robin, in the Eyrie.
When the war of the 5 kings breaks out, Lysa is too fearful and paranoid to get involved, believing she'll be safe if she never leaves the walls of the Eryie, and keeps her army out of the fighting and nearby to protect her.
As the war of the 5 kings comes to a close, Peter Baelish makes his move to gain power by marrying and quickly murdering Lysa, thus leaving weak Robin as Lord of the Eyrie.
Robin is an underdeveloped, winy, child, and has little to no understanding of what's going outside of the Eryie except for what information his mentally ill mother has fed him, or what Peter Baelish told him. This leaves Peter with the power to control which battles the Eryie gets involved in. For the most part Peter sees no gain for him to get involved in any battles until the battle for the North. Which he does in hopes of regaining the trust of Sansa, and thus having a key ally in the North.