Thanks for writing to the forum!
It is difficult to comment without examining. The rash that does not itch could be actually petechiae. Petechiae are pinpoint-sized red dots under the surface of the skin.
The commonest cause of petechiae is a low platelet count (thrombocytopenia).
They can also result from autoimmune disorders, such as lupus or rheumatoid arthritis, bone marrow disorders such as leukemia, inflammation of the veins, as a side effect of certain medications including the vaccine itself, viral infections, such as EBV and septicemia or infection in blood. A complete blood count, urine examination and a blood culture may be required to reach a diagnosis. Elisa test will not help in this case. This is conclusive only if there is some allergy.
Hope this helps. I think you should discuss these possibilities with your friend’s doctor. It is difficult to comment beyond this without examining. A comprehensive investigation is required keeping all the points in mind. Please let me know if there is any thing else and do keep me posted. Take care!