I just recently bought a pc that comes with an ATI Radeon HD 5450 video card. I'm wondering if this graphics card would be able to run games like Final Fantasy XIV at high resolution without any issues with frame rate.
Doubtful. A 5450 is an entry-level card. While it could play the game, it won't do so very well at high resolution without dropping the detail levels significantly.
Take a look at the reference chart here comparing cards from both nVidia and AMD/ATI: http://www.tomshardware.com/reviews/geforce-gtx-460-rad...
Note that the 2900-series cards are more powerful than a 5450. (5450 is actually quite low on the list, whereas a 2900XT is much closer to the top. Use the "Find" function of your browser to look for them.)
Cards like the 5670 and above would be better suited for the task, but you may even want to go a bit further to a 5750 or 5770 to get greater detail out of that game.