Despite excellent grades, does having no rapport with any member of the faculty preclude my getting LoRs?
Yes, you can ask for recommendations from people you don't know well; send an email with one to three of your best papers attached, as well as an unofficial transcript; in the body of the email, remind the professor which course(s) you've taken with that person.  Include a sentence or a paragraph about what draws you to the idea of grad school, that is, show your spark.
Yes, you can apply for grad school with mediocre letters; you may or may not get in.  (I do not know whether the letters you would harvest at this point would be mediocre or not.)  You are right that this might not be the optimal way of proceeding.
Now I'll address what you asked in a comment.

&quot;What does being more involved look like?&quot;  In other words, as a student with Asperger's, who is not naturally outgoing or extroverted, how can I get to know my professors better, with the goal of garnering more effective letters of recommendation?

I'm glad you asked this well in advance of application season, because this gives you time to get to know potential letter writers before you really need the letters.

Ask questions and contribute to class discussions.
(You've done this -- check!)

Visit office hours to start to build an academic relationship.
You don't have to need help with a project in order to visit office hours.  If you are not sure how to structure a conversation in office hours, you could send an email requesting an appointment (so that the professor isn't thrown off thinking you're just coming in for five minutes to, for example, request clarification of an assignment); if I were in your shoes I would say, quite clearly,



&quot;I would like to go to grad school and am hoping to get to know you better, so that I can comfortably ask you to write a letter of recommendation.  However, I have Asperger's and am not accustomed to visiting office hours, but I would like to try; I may need some help getting started with having a productive conversation.&quot; (You can use this as is or modify it to make it your own.)


You can also start a conversation through email.  Some individuals are more comfortable with email than in-person visits.  If you're not sure which would work better for you, you might want to try both approaches.
In office hours and in email correspondence, in order to break the ice sometimes it is helpful to share a list of universities you are thinking about applying to, and ask for help in narrowing down the list, or adding to it (depending on the length of your list!).  In other words, you can ostensibly be asking one thing, as a way of getting the ball rolling.

Note that you can ask your university's office of students with disabilities for assistance if you want to.  There is no guarantee that they will help, but they might, and as long as you don't put all your hope in that external assistance, there's no harm in trying.  (Some of these offices are great, some less so.)

You might find it helpful to have a grad-application buddy.  Sort of a partner in crime.  Someone who's going through the same process as you, and you support each other through the process.  In fact, your application buddy might go with you to the first couple of office hours visits for moral support.