The SVI is simply a function (empirically fit to the data) which given a maturity and a strike price K, computes a BS implied volatility $\sigma$. Once you have that implied volatility you can plug it into a Black Scholes routine which can compute the BS price and the Black Scholes greeks.
Note that if an option is actually traded with that strike and maturity you could have directly observed the price, and computed the volatility and greeks directly from that.
So the SVI technique helps compute Greeks in 2 circumstances: (1) if there is no option traded for the strike and maturity you have in mind (you could call this a "what if" calculation of the greeks), (2) if you think the price of the option is somehow noisy or distorted, in which case the SVI based calculation might be more accurate because it is fit to multiple options, not just the one you are interested in. However IMHO the market prices of options are pretty accurate.