Your question did not specify to which stage of aveilut you are referring. We will begin our discussion with the most severe stage – shiva.
The Maharil[1] says that one may not give mishloach manot to a mourner during the year of his aveilut for a parent, because doing so is considered like sh’eilat shalom (inquiring about the mourner’s welfare), which is forbidden during this period.[2] If giving mishloach manot to an aveil is equated with sh’eilat shalom, it follows that the mourner’s giving mishloach manot to someone else is like his sh’eilat shalom to that other person. The halacha is that a mourner may not do sh’eilat shalom to other people during his shiva, whereas it is permitted thereafter.[3] Thus, it would seem logical to forbid an aveil from giving mishloach manot during shiva.
Nevertheless, the Shulchan Aruch[4] rules that an aveil does give mishloach manot during shiva, because even during this period he is required to fulfill mitzvot, including this one.[5] (Note that the Magen Avraham[6] writes that such mishloach manot should not include special “treats” that evoke particular delight when received.) However, an aveil during shiva is forbidden to give presents if they are not mitzvot that must be performed during that specific period.
As we mentioned, after the shiva is completed, an aveil is allowed to do sh’eilat shalom to others. It is therefore reasonable to assume that the aveil is allowed to give presents at that point. Although some Acharonim forbid a mourner for a parent to give presents throughout the entire year of aveilut,[7] this stringency does not seem to have a strong basis. Even if one wants to be stringent on the matter, it seems clear that this is only when the present is conspicuous in its ceremoniousness, but not when the nature of the gift or the relationship between the mourner/giver and the recipient make the gift a matter of course.[8] Only a festive type of gift would parallel to the Magen Avraham’s ruling that an aveil should not give the type of mishloach manot that evokes special joy.
Let us turn our attention to your question regarding giving tzedaka. It is very appropriate for a mourner to give tzedaka throughout the various stages of mourning, both for the merit of the aveil[9] and for that of the deceased.[10] Indeed, it is difficult to find a good argument why an aveil should not give tzedaka after the shiva is over. Although giving tzedaka should bring the donor joy, the joy of doing a mitzva is not something the Rabbis prohibited for a mourner; only actions that are joyful by their very nature, irrespective of their religious/moral content, are problematic. Furthermore, if one gives money to a tzedaka organization, it is unlikely that there will be any interpersonal interaction that might even raise a question of simcha or sh’eilat shalom.
Even during shiva, the aveil is obligated to fulfill mitzvot like anyone else, and this includes giving tzedaka. Indeed, tzedaka is considered a good way to bring merit for the deceased. As the Mishna Berura[11] comments, if a mourner during shiva is obligated to give mishloach manot, he certainly should give matanot la’evyonim. The same applies to giving other tzedaka. The only difference is that matanot la’evyonim must be given on a specific day, whereas some donations of tzedaka can just as well be given afterward. Although we would not recommend that a mourner during shiva occupy himself with the periodic writing of tzedaka checks or the like, if a particular need arises (e.g., a collector comes to the door or there is some pressing need, etc.), the mourner is not precluded from donating appropriately. In addition, we might suggest that shiva is a good time to decide to give significant donations, whereas the actions of actually giving the money would be carried out after the shiva is over.
There is one tzedaka-related limitation that we do suggest a mourner observe throughout the year of aveilut. Large donors are periodically honored at dinners and in other ways. While the year of aveilut is an appropriate time to donate enough to be honored, it would seem inappropriate for the donor to accept the accompanying festive honors during this period.