My husband has Chabad roots and the Chabad minhag is for a Bar Mitzvah boy to put on Tefilin 2 months before his Bar Mitzvah. Our son is turning Bar Mitzvah in Tammuz so our original plan was for him to put on Tefilin this month in Iyar but now that shuls are closed and we weren’t even able to buy him a hat yet should we wait a month and do it one month before his actual Bar Mitzvah and hope shuls will be open by then? If we do it now would we be able to buy him a hat even though its Sefira?
If your minhag is to start chinuch to put on tefillin two months before the bar mitzvah, you should stick to it, even if he will not be able to do it in a shul. The kiddush that is given in honor of his putting on tefillin can also be done when the opportunity presents itself, even if it isn’t the first day that he is putting on his tefillin. It will be a celebration for the opportunity to put on tefillin, (everyday that we can put on tefillin is reason to celebrate!) be If you can get him a hat that is great, but even if not, he can put them on even without a hat, (by the way, Bar Mitzva boys in Eretz Yisroel don’t put on a hat until the day of the actual Bar Mitzvah).
Regarding buying a hat during Sefira, according to most poskim, it is permitted to buy and wear a new garment during sefira.
Ma’amar Mordichai 493-2, M:B 493-2, Kaf Hachayim 493-4, Likutei Maharich, Emes L’yackov 493 ftnt. 466, KOvetz Halachos ( Sefiras Haomer) 8-5.