Actually there are some automated fertilizer injector systems available on the market already, albeit at a fairly high cost. Yes, you can certainly use a syringe for injecting liquid fertilizers into the plants, but you have LOTS of obstacles to overcome:
1) You have to devise a way to push the syringe at a very specific (low) rate (i.e. gears are necessary plus other parts in combination with a stepper motor)
2) You have to make the formulations yourself while ensuring that all the ingredients (usually various salts) get completely dissolved in the water and don't precipitate out later (especially iron compounds which are quite prone to this)
3) Special measures need to be taken to prevent mold/bacterial/algae growth in the syring because the nutrients (plus the water) will almost certainly attract them for sure
Thus once you know how to take care about all the aforementioned issues (and then some) you're all set.