This sounds totally off the wall to me on so many levels. I agree with others that I typically hear EMDR is riskier and potentially less effective for those with complex or childhood trauma.
The fact that you are interpreting this relationship's issues as "possibly my fault" is a giant red flag to me. I don't think clients should feel that way if their T seems to be staunchly attached to how things should be or need to be. It sounds to me like you are trying to compromise with a T who is unwilling to meet you halfway, and then you're turning around and thinking about maybe blaming yourself for not bending her way? I don't know what social trauma means, exactly (Domestic violence? Family violence? Cult environment/totalitarian regime?), but it sounds like self-blame could easily be a resulting symptom of such trauma.
It's also a huge red flag to me that she is telling you there is a memory you need to uncover somewhere. That is so wrong to me. If she said she feels like there might be early trauma you haven't come to terms with, that's one thing, but saying, "No, this isn't the right memory. Find me another from this early period," THAT sounds like a recipe for recovering false memories. Wtf? Is anyone else hearing this risk?
I think you should call around to other Ts who do EMDR for trauma, Ts who will give you a free phone consultation. They don't have to be viably in your area. Call a few that look good, wherever they may be geographically, say you are interested in their services, and then quizz them on your current EMDR experience. If you find some good ones, I would hope they wouldn't just be looking to score a client and may give you some frank advice/opinions/insight on/into what your current T is doing.
I think this whole shebang sounds like your T is playing with psychological fire. She sounds dogmatic. Dogmatism does not seem particularly conducive to trauma therapy IMO. And I have never heard someone say that EMDR is exclusively for people with early trauma. Regardless, if EMDR were exclusively for people with early trauma (which it totally is not), then she should tell you to look into alternate modalities, not to find her an acceptable trauma to delve into.
I just seriously cannot believe what I am reading. Personally, I find this treatment outrageous.