As far as I can tell, Dragon City is a Flash game, so you shouldn't have any trouble playing it on a 32bit Windows Vista. Additionally, the game is available on Facebook, so you don't even need to install it on your computer.
Although your browser might be equipped with Adobe Flash Player, I strongly recommend to update it to the latest version through the available website. This eliminates possible problems that you might encounter from the start.
Here is the link you have to visit, and based on your browser's name, download the correct package and install it.
Note: Latest version of Adobe Flash Player is fully compatible with Windows Vista 32bit.