Of course you can tap into any digital line, and unless it's got some undocumented encryption you can decode it. But if your intent is to 'make a different kind of guitar tuner' then you learn nothing about that and gain nothing other than learning how to decode an LCD.
A tuner has 3 aspects, input, process, output. It -can- be done in 1 chip, so there's no 'reading the data being inputed into the tuner, instead of reading the data outputed to the LCD.' At least, that's the cheapest way to do it. Regardless, if there's a separate analog input interface chip, and even another driver to the LCD, there's no other point where you can tap into 'data.'
You'll learn a lot more and make more -direct- progress towards a tuner by researching and studying waveform analysis. There's got to be a ton of info available about it ranging from algorithms-101 tutorials to whole books. Start with the -101. 
'I need really good frequency detection so I don't think that if I made one myself (including coding the freq. detect.) it would be accurate enough.'
You're not going to get that by decoding a $20 tuner either. A consumer grade item is designed/calibrated by referencing against lab grade tools. There are also lots of hobby grade versions too that can get you off the ground. The 'accurate enough' is in the ADC resolution bits. That's it. After that the software is researchable. Check sourceforge.net. I'd half-bet there's an open source project or even several projects going on there. Check github too. Now those are more likely about PC based tools not Arduino, but they'll point your researches in the required directions.