I think Dental Fear Central could be really helpful in finding a nice dentist/helping to tackle your daughter’s phobia - but I also think she needs to do those things before embarking on orthodontic work.
As I said above, I am not scared of the dentist, have never missed a checkup, and actually had a load of orthodontic work in my teens so my teeth look great. But, I am absolutely terrified of needles! I pass out at blood tests, and cry at my flu jab.
When I had my “bad year” I basically had to get past that phobia because I needed several root canals, and one tooth extracted. The alternative would have been sedation and them all whipped out (probably by someone I didn’t know because my surgery doesn’t offer sedation). Then I would have been left with a load of missing back teeth.
It was only possible because I have a lovely, patient, kind dentist, who I have been seeing for a long time. I wept in the surgery many times, proper ugly crying (i’m In my mid 40s!) and she was nothing but supportive. She might gave thought I was a pain, but she never once showed it. The dental nurse held my hand through every injection, we took things at my pace (longer appointments than I really needed) and I developed my own coping strategies. Some of these were completely ridiculous - I took a soft toy to every procedure so I had something to hold in my hands, and I bought myself a “present” after each treatment was finished. My dentist probably thought I was bonkers, but she was with me every step of the way.
I honestly think finding a dentist who would eg just chat to your DD, build up to having a look in her mouth in a normal chair, tiny baby steps, is the only way forward. Orthodontic treatment would be down the line from that. It probably sounds daft, but during that period I felt like my dentist, the nurse, and I were “a team”, working together - I think that’s where your daughter needs to get to even though it might be a long and difficult road. (I just wanted to keep my teeth, she has the goal of lovely looking teeth!)