Is it possible to be allergic to the yolk of an egg but not the egg white?
The protein composition of the egg white and egg yolk differ appreciably. Hence, it is plausible for one to countenance symptoms of indigestion when consuming the egg yolk as opposed to the albumen, egg whites. It is important to stress that your friend may only an indigestion towards egg yolks as opposed to an immune response to an antigen. 

If we consider the protein family, the egg yolk comprises a family of phosvitins, a group of highly phosphorylated proteins capable of mustering iron and calcium metal cations - serving a role in embryo development.

These are primarily absent from the albumen which comprises (with an unexpected name!) a highest percentage of suspended ovalbumin, ovotransferrin and ovomucoid. As demonstrated by the American College of Allergy, Asthma &amp; Immunology, it is feasible to suffer an intolerance towards the egg yolk and yet suffer no gastro-enteric indigestion to the egg whites.
More information regarding the American College is available from: https://acaai.org/allergies/types/food-allergies/types-food-allergy/egg-allergy

Your friend evidently does not suffer a severe anaphylactic shock after consumption of the egg yolks otherwise he/she may be adverse to both the egg whites and egg yolk. In conclusion, it is possible that he/she may simply have a mild intolerance towards certain proteins in the egg yolks.