Been here and done that with my son. Yes, could be a form of social anxiety. Unfortunately, at her age, there is no reasoning with her. All you can do is talk to her (the more you do, the more she will understand as she ages.) and try to get her ready for each "change". Of course, it may not be social anxiety, but Sensory Intregration Dysfunction. There are two kinds, hyper and hypo. One kind the child goes through extremes to feel things, like hitting themselves or running into walls; the other, they can't deal with stimulation like loud noises, textures, lots of people, etc. The kids almost appear Autistic because when they can't process the environment properly, they either withdraw or cry.
I truly believe that more you socialize her and gently introduce her to the overwhelming world, you will slowly see changes. My son was EXACTLY like this. We couldn't take him to a parade, watch fireworks. I couldn't vaccuum or use the mixer without him HYSTERICAL. He cried everytime the lawn service came to cut the grass. I just told him I (or we) had to go somewhere or I had to clean. I then told him how to handle it (wear your earplugs; go in your room and play with your door closed, etc.) As he aged, it gave him a sense of control, and as he got older, he became more curious, and would actually watch me vaccuum, or want to go to the parade (though the noise and people were hard for him, he liked the trucks and floats, so he would be brave.)
I never took him to a doctor about it, though I did let his ped know, as well as know his progress (or lack of). I had read an article about Sensory Intregration Dysfunction, then borrowed some library books on it, and that really helped. At 4, he was diagnosed with ADHD (which I still don't believe, as I've had other professionals over the years tell me it's the "easy" diagnosis some doctors give, and as he's gotten older, he doesn't act at all like a "ADHD kid".) and borderline Autism. Now that he's 8, it's much more apparent that he's Obsessive-Compulsive, which has some shared behaviors with ADHD and Autism, as well as other social anxiety disorders, which OCD is.
I dealt with this issue for probably 3-4 years before getting true progress, which was when he hit 3 1/2 to 4 years old. That's when he went to speech, and a part-time preschool. I would NOT put your daughter into daycare BEFORE taking her to a developmental psychologist. The stress will be very very hard on her, as well as the daycare teachers and even the other daycare kids who will have to listen to her cry for the entire day. But please know that her behavior will not stay this way forever, with patience and love on your part. My son is a polite person that shakes people's hands and is just fine at parades and parties now, though he still has issues making friends and needs reminding on the right social behaviors. Good luck! I hope your road isn't as long as mine was.