How to get recommendation letters for graduate admissions when my social anxiety makes it difficult for me to make connections? I understand references make a huge difference when applying to graduate programs. 

Over my years in university, I have spoken to professors about projects / exams / papers. I definitely make use of their office hours.  I also go to a top school and am set to graduate with high honors. I am an extremely determined and motivated person. Although I have made use of the available resources, this is an extremely difficult/rigorous school and thus very few people achieve anywhere near a 4.0. My cGPA is a 3.5 and my GPA over the last 2 years is a 3.7. I have also been involved in a lot of organizations related to my field of interest/research. Including, volunteer on three projects at a renown research hospital and research for an undergraduate science magazine. 

However, I have severe social anxiety. I understand this is “no excuse” in academia for not making connections, but it is extremely difficult for me to be around other people. It is severe. It has been a huge step for me to even go to the office hours (which I am proud of myself for doing), but I have not stood out from the crowd and I doubt any remember my name. I have never wanted anything more (graduate school), and yet my mental illness is a severe restriction on my life. I have been seeing a psychiatrist and have made huge improvements since high school (when the social anxiety became severe).

Is it necessary that I only acquire references from professors with whom I have developed a close relationship? Should I discuss these struggles in my application?
I can see this is a difficult spot for you.  After all, to be good at your work doesn't necessarily require you to love interacting with people, but to get into the field you require letters of recommendation, which are best achieved by... interacting with people.

A few thoughts:


Are you more comfortable communicating over email?  If you can combine a good email correspondence with good course work, I think there's a path to overcoming this.  You can easily email your professors (using the less verbose style of your revised OP!) and ask about graduate school, research interests, and finally letters of recommendation.  
Have you taken classes with the same professors repeatedly?  This is one piece of advice I give to undergraduates a lot.  Each good semester with a professor helps reinforce the positive impression.  Assuming you still have one semester left before finishing, I would prioritize doing this again.  Speaking for myself, student's names I only see once are often quickly forgotten unless I've gotten to know them in person well, but student's names I see repeatedly start to stand out regardless, especially if they've done solid work.
Related to 2, have you taken only very large classes with the professors you want letters from?  If so, and if it's possible, you might try to get into a smaller section they're teaching even if it's of less (but not unrelated) interest to you academically.
You would be surprised how far showing up to office hours with good questions will get you, I think.  You said you've managed to overcome your anxiety enough to do this, and I don't think you should underestimate the benefit.  


In summary, find alternative ways to build relationships with the professors, through repeated interactions and electronic communication.  You don't have to be chumming it up regularly with your letter writers in person to get good letters of recommendation, though even the few visits to office hours you have managed will have likely helped.  Also make sure you read the posts others have linked in the comments regarding mental health issues and PhD admissions.

Good luck!