The Family Med dock that I shadowed had me wear one of his white coats. He just had me cover up his embroidered name on it with a couple of layers of medical tape so his name (and MD) did not show through. Not that any patient would confuse me with "Attending Doc, MD" ha ha.
Next week I start a "summer observership" at the school where I am matriculating. The attending requested that I bring my "white coat" to wear. The school white coat ceremony won't be until the middle of August, and the only white coat I have is from a chemistry classroom that is covered in various chemical stains (silver nitrate, iodine, a little nitric acid, and maybe some biological stains.), burn marks, and marks from leaking pen ink. Probably not too appropriate for a clinical setting. My school came to the rescue because they said that they have white coats to loan out for these circumstances.
As for "formal" attire, maybe "business" attire would be a better way to phrase it. In my mind, "formal" means a tuxedo/black suit or a prom-style dress. On the other hand, "business" means a button/collar shirt and tie with slacks, or the female equivalent.
As for your "white coat" dilemma, go to a uniform store, or if you have time you can shop on-line. You can get a short white coat in your size if you don't already have a clean one from your science classes in school.
Good luck. I hope your shadowing experience is good.
dsoz