My understanding is that EMDR is something that can be applied if you have something specific to hold on to - a memory, an intense emotion, something. I don't have much belief it works with severe avoidance because there is nothing to work with. That not only applies to EMDR but also to any kind of modality. The T must be very patient, alert and attuned to find something that accidentally slips through. I wouldn't blame the avoidant person though. The avoidance is there for a reason and the core of the work is establishing enough trust to lift this avoidance even a tiny bit.

I've once asked if EMDR could be something that could be useful for me. The answer is no - I don't have any specific traumatic memories. My whole childhood was emotionally neglectful but I did not perceive it so as this was my normal. So it would be impossible to find anything for the EMDR to focus on.