Accessing old games in my playstation plus subscription
Solution 1:
You will only be able to download the games from past PS Plus months if you specifically claimed the games during the month they were available. You can claim games you don't actually have the console for, but you do have to claim them all individually to add them to your download list.
If you added the Vita games by completing the checkout in the Playstation store every month, then they are already in your download list and you can use them on your new Vita.
Any games you didn't claim in the month they were available can not be added anymore.
As games from other platforms may not show up on your console store (e.g. PS3 plus games don't show up in the PS4 store), you can use the PC store to claim games on all platforms, including ones you don't yet own, at store.playstation.com. Here, you can also view your download list and see which Vita titles you already have. You can download all Vita games that show up in that list once you have the platform.