A few comments:
–You would be severely limiting yourself if you only apply to colleges that don’t require recommendations. There are very few (Penn State is one I think).
–You are not the only quiet student who has gone through your HS. Teachers can still write you an overall positive LOR that can focus on your academic achievements, preparedness for class, kind treatment of other students etc. even if they choose to note that you are quiet in class. Think about what teachers you particularly like, what classes you excelled at and ask them if they will write you a positive LOR… If they say yes, thank them and ask if there is anything you can provide (ex. resume, list of activities) to help them write the letter.
–Please be realistic in the schools you apply to in terms of academics and affordability. Agree with @Lindagaf that while you have done very well in HS, your stats don’t put you in the very top tier college range. Your price constraints makes your in-state public a good option unless you find schools where you would qualify for need based (you need to run net price calculators) or merit aid.
–FWIW my S was very quiet in HS and got what must have been two perfectly nice LORs as he got into his top choice colleges.