Can platelet rich plasma cause an allergic reaction in the body?

Unlikely. The platelets are your own and should not cause an allergic reaction. The injection will cause a local inflammatory relation that will be painful for 4-6 wks at the injection site.
No. Platelet rich plasma (prp) is a component of your own blood so it is unlikely that you would be allergic to something that came out of you - it's a part of you. Prp injections are meant to stir up healing and the first stage is an inflammatory response, so its not unusual to have a natural inflammatory reaction characterized by pain and swelling, but this isn't an allergic reaction.
Not "allergic", but. ..immune? Yes. So I humbly disagree with the previous answering docs. I personally had a patient have a flare after an orthopedic spine procedure. Patient eventually was found to have previously undiagnosed Psoriasis. Good news is the patient still did great, even coming back for more treatment. Most of my experienced regenerative medicine colleagues have a few similar stories.