Update: Like I thought, Vitamin A injections in the arm are dangerous just like an IV. Here is the reply I received from a doctor at a nutritional IV center:
"Fat-soluble vitamins must be absorbed through the GI tract. The place of
injection does not change the fact that fat soluble vitamins should not be administered by any means other than by mouth.
This is why you will not find anyone with proper training and credentials providing these vitamins by IV or intramuscularly. There is one
exception and that is Vitamin K which can be administered by injection or IV for the emergency reversal of Warfarin therapy."