From Moscone to the Dragon's Gate it's about a 10-15 minute walk depending on how fast you go. [
MAP]. Grant and Stockton next to it run parallel up through Chinatown up to North Beach. Grant's the touristy one (lots of shops) and Stockton has a lot more locals and markets.
As for vegetarian options close to Moscone, the two that come to mind are
Samovar Tea Lounge right there at Yerba Buena Gardens, or
Loving Hut in the Westfield Mall which is also a stone's throw. Definitely Love the Hut if you're trying to save money. Westfield has a number of other food options as well.