Would it be wrong or cause any harm for me to pray with a traditional Jewish prayer shawl like the one Jesus and the Jewish apostles wore? It would be in private prayer only (with the door shut) so not to cause confusion to others or draw unnecessary attention to the item. I would ask my priest bless it as well. I find our Jewish roots so interesting and feel like all the meaning and history associated with the prayer shawl would enrich my prayer life.
From the phrasing of your question, I take it that you are not a convert from Judaism and are not of Jewish heritage. In that case, the answer is: No, you may not. A Jewish prayer shawl (Hebrew, tallit), is a sacred liturgical vestment for Jews. Many Jews feel very strongly that wearing a tallit, like wearing tefillin (phylacteries), is a privilege reserved to Jews. For a non-Jew to presume to wear one for personal prayer would be similar to a non-Christian Jew presuming to obtain a priest’s chasuble to use for his personal davening (Jewish prayer).