Is there a department of philosophy at your current institution? Are there any formal/semi-formal ways that professors and graduate or undergraduate students in philosophy meet as a group? Such as a seminar series or colloquium? Assuming you are applying for the fall where deadlines are often somewhere around December 1st, you still have time to participate in these. If you don't know where to find out about these sorts of events, it would be appropriate for you to ask one of your professors, especially if you state your future interest in graduate school. "I am interested in graduate school in philosophy and I was wondering if there are seminars that graduate students here participate in, and if so, if I can join as well."
Understandably, these might be uncomfortable situations for you, but perhaps your shared interest in philosophy would help distract you from the social aspects (especially because it sounds like you successfully cope in lecture settings, and have even been comfortable asking questions, which might put you ahead of many fellow students who would otherwise identify as neurotypical).
Otherwise, or in addition, how do you prefer to express yourself? It sounds like you have some success with writing. I think you could also write to a current/former professor who shares some of your interests: "I am interested in topic XXX and I am interested in a future in graduate school in philosophy. I dislike some social situations because of Asperger Syndrome but I am interested in discussing the topic with you. Would you have time and be willing to read and comment on some writing I intend to do on the subject outside class (~xx pages)?"
Professors may or may not have time to accommodate you with that request, but at a minimum it shows that you have some interest in the topic beyond coursework which won't hurt you, and in the optimal situation, you will interact with that person more closely and have an opportunity both to express your interests and abilities and to have a contact person for suggestions for other aspects of the graduate application process (such as particular programs/professors at other institutions that will also align with your interests).
And just to add a response to your actual title question: No I do not think it precludes you getting recommendation letters, but you will probably not get the one stellar letter that makes an admission committee look extra close. Depending on how the rest of your application looks as well as the personal opinions of the admissions committee, it might not matter at all in any case. Recommendation letters can be difficult to interpret. Given the time you have between now and when applications are likely due, I would advise that you make some effort or set a goal to have at least 1 professor who can give you a more personal letter of recommendation, through some of the strategies I suggested as well as @aparante001. Even if you don't feel like you are able to accomplish this goal, I would suggest you apply anyways with whatever letters you are able to obtain. If your coursework is as strong as your impressions are, I think those letters will read much better than mediocre, they just won't be intensely personal.