Can't Download the App my phone is old

Hi. Most likely YES. This new apps are working with the new software. They usually support IOS7,8,9. But IOS 6 is becoming Obsolete. A good advice, I use to have the Ipad 2.You can update your Ipad to IOS7. It will work good, you will not see the difference.

You don't say what model iPhone it is, but the compatible devices for the current version of Vine (from the app's description page in the store) :

Compatibility: Requires iOS 8.0 or later. Compatible with iPhone 5s, iPhone 6, iPhone 6 Plus, iPhone 6s, iPhone 6s Plus, iPad Air, iPad Air Wi-Fi + Cellular, iPad mini 2, iPad mini 2 Wi-Fi + Cellular, iPad Air 2, iPad Air 2 Wi-Fi + Cellular, iPad mini 3, iPad mini 3 Wi-Fi + Cellular, iPad mini 4, iPad mini 4 Wi-Fi + Cellular, iPad Pro, iPad Pro Wi-Fi + Cellular, and iPod touch (6th generation).

So it won't be compatible with your iPad 2 either.

(The previous reply about updating to iOS 7 is incorrect, you can only update a device to the highest iOS version that it supports, which for an iPad 2 is the current version i.e. 9.1, you can't update to an intermediate version - but as the developers Vine haven't made it compatible with an iPad 2 you still won't be able to download that app even you update your iPad.)