Yes, it is possible determine by skin tests sensitization both to vaccines and the potential allergens present among the components of the vaccines' formulation.
I should disagree with María Luisa Caballero, I am afraid. Although in principle skin testing may be helpful, the sensitivity and specificity of skin tests in predicting SARS-CoV-2 vaccine reactions remain unknown. Thus, it is not suitable to use this approach in routine clinical practice.
Allergic reactions can be due to allergy to vaccine ... Immediate hypersensitivity or allergic reactions to ... A prick skin test is performed with the vaccine diluted 1:10, and if ... https://waojournal.biomedcentral.com/articles/10.1186/s40413-016-0120-5