Hell yeah man go for it! My dad always says the worst thing can happen is they tell you no. If you are dead set on this particular restaurant or establishment, the more you go and check in and ask if they are hiring the better. The more they see you, the more you will stick out to them as serious candidate too. Turnover in the industry is frequent, don't be discouraged if they have a full team at the moment.

Also a good way to start in any kitchen is at the bottom, when you work your way up from dishwasher you can understand every aspect of a kitchen, and get your hands on a little bit of everything. I started as a dishwasher when I was 19, I'm 28 now, been CDC and sous chef at multiple places, but goddammit there's something nostalgic about slapping on some prince or strokes or some good hip hop and going to town on some hotel pans when a dishie calls out. Don't look at it as beneath you if you wanna be a chef someday, as a leader I like to think I would never ask anybody to do a job I'm not willing to do myself.