There is no prohibition for an Avel to give gifts or to receive them. The Ashkenazim are stringent not to send a gift to an Avel until the end of the year however he is allowed to give gifts.
Explanation:
The Gemara in Moed Katan 15a says: An Avel is forbidden in She’ailat Shalom-literally asking Shalom.”
The Gemara on 21b says: The Rabbis taught an Avel is forbidden in the first three days of Aveilut in She’ailat Shalom. From three days till seven he can answer someone asking but he can’t ask after that he may ask or answer like regular.
The Shulchan Aruch (YOD 385:1) writes: An Avel is forbidden in She’ailat Shalom. When? The first three days he can’t ask Shalom, and if others don’t realize he is an Avel and they ask him his well-being, he can’t answer them, just tell them he is an Avel. From three till seven days he can’t ask, but otherwise who don’t know he is an Avel he may answer them. From seven days till the end of Shloshim eh may inquire about others since they live in Shalom, however, others should not inquire how he is doing. Of course, if one asks him who doesn’t know, he can answer them. After Sheloshim he is like everyone else. That is by other relatives but by parents, he may inquire about others after Shiva but no one should inquire about his well-being until after the end of twelve months.
The Rema writes: There are those who are lenient nowadays in She’ailt Shalom of an Avel after thirty days but there is no basis for this unless they understand that asking about a person is not what the Gemara means by She’ailat Shalom.
The Shach asks if that was true then we can do our She’ailat shalom even during thirty days.
The Be’er Heitev answers the Shach’s Maybe we can answer that since most She’ailat Shalom we have nowadays just “Good Morning” which is permitted because it is not actual She’ailat Shalom, (as defined in as OCH 89:2). The Bet Yosef quotes the Ri saying Shalom with an action like bowing is the problem but saying Shalom or any greeting is fine.
There seems to be a disagreement between the Shulchan Aruch and Rema if is permitted to Greet an Ave after sheloshim.
Regarding gifts which are an outgrowth of She’ailat Shalom_
Shulchan Aruch 385:3 writes: In a place where the custom is to ask Avelim Shalom on Shabbat they can do it. The Rambam writes the Avel should greet everyone on Shabbat since if he doesn’t it is as if he is showing Aveilut publicly on Shabbat which’s forbidden.
The Rema writes: its forbidden send gifts to an Avel on his parents for twelve months, even on Shabbat in a place where the custom is not to greet him on Shabbat but in a place where it is then permitted for a gift.
The Rema is talking about sending packages and gifts to others but the Shulchan Aruch is not discussing sending gifts at all. according to the Shulchan Aruch, there is no prohibition on greeting not sending gifts.
So, understood the Yalkut Yosef Hilchot Aveilut103:16, “An Avel during the year of his parents who get married can receive gifts, even if he is prohibited in She’ailat Shalom the is no prohibit on giving gifts. Especially money to help him get set up.
Halacha 17,” A Kallah who is in Aveilut during the twelve months for a parent is permitted for the Chatan to send gifts before the wedding.
This same machloket is seen in Purim as well. The Shulchan Aruch OH696:6 writes an Ale is obligated to send Mishloach Manot. The Rema qualifies, however, no one should send him during the twelve months.
