Traditionally, pre-Reform, Conservative* et al Judaism, the "Four Species," Arba Minim, lulav, hadasim, aravot and the etrog, was male only. They did the shopping, rather pedantically, for them, and they took their set to synagogue, waving them and marching around, while women watched or stayed home.
Vayikra (23:40): “And you shall take for yourselves on the first day [of Sukkot]1 the fruit of the hadar tree, date palm fronds, a branch of a braided tree and willows of the brook.”In some families the women, in the privacy of their homes, also made the blessing and shook the four together, but that was not a public act.
One thing I've noticed in recent years is that it is becoming routine for there to be a few women in my neighborhood synagogue with their own sets which they bless and shake up and down, just like the men. Due to a lack of space, marching around would be impossible, so it's hard to see just how many women come with their own set. I must admit that there still aren't many, but those who do bring them are from the "best" most distinguished families in the community. These are not the families with rebellious children and "datlashim," the slang (or euphemism) for those no longer religious.
I am not an expert in Jewish Law, but from the little I know, this is certainly a mitzvah that women can do without causing any halachik (Jewish Law) problem. It doesn't at all interfere with the prayers of anyone else nor affect any other aspect of observance. Women are certainly permitted to have their own set. It seems to me that this trend will grow, like the number of women who take it upon themselves to say Kaddish with a minyan for the year after the death of a parent.
*I do not know if it is the norm in Conservative and Reform communities for people to have their own personal/private sets of "Four Species," Arba Minim. But I have no doubt that in the "Egalitarian" or "Open" Orthodox communities there are many women with their own sets.