Neural networks are powerful, but they require tremendous compute resources and a large training set to be used effectively. For a lot of cases they’re probably the best tool available, but I regard them as rather a sledgehammer solution - they’re powerful and get a lot done, but lack the finesse and efficiency of a more tailored solution.
Regarding the original problem, I’m going to stop right at the title. 100% accuracy is impossible. Period. Can’t be done. Humans are much, much, much better at reading handwriting than computers are, but it’s still not hard to find handwriting samples that are virtually or completely illegable to a number of people. (I probably have a less than 50% accuracy rate at reading doctors’ handwritten notes, and I use cursive handwriting far more than printed handwriting.)
Whenever you’re looking at something like image processing/recognition, you’re guaranteed a lower than 100% success rate, because even a number of tasks that humans can perform intuitively (and far better than the best machine implementations) have a non-100% success rate. Making out written letters is not easy when it’s anything other than a printed page, and even that has issues (if you can read a page with 100% accuracy, switch the font - sooner or later you’ll find a font that’ll screw up your recognizer). You’ll be best off using an existing service like the others have suggested.