There are many precedents for women’s readings of Eikha and kinot (the mournful elegies that are recited) on Tisha B’Av. This is apparent from numerous sources including the medieval Kol Bo, which describes the laws and practices of Tisha B’Av.
And therefore they extinguish all the lamps and the cantor begins to recite the lamentations to sadden the souls and break the hearts. And they recite dirges there for about quarter of the night, the men in their synagogue and the women in their synagogue. And likewise during the day the men recite dirges by themselves and the women by themselves, until about a third of the day is past.
(Hilkhot Tisha B’Av, 62)
There is no stated halakhic obligation to read Eikha and therefore no issue of needing someone else to read on behalf of others. The source for our practice of reading Eikha on Tisha B’Av is Tractate Sofrim (14.3), but this is essentially a gaonic work and the status of reading Eikha is a custom and not absolute law.
Obviously the issues of kol isha and kavod hatzibur can be used to exclude a mixed reading of Eikha, but for congregations for which this is not seen as a barrier, a mixed reading of Eikha would be acceptable. Although the Vilna Gaon advocated reading from a parchment scroll with a berakha, (which is what Tractate Sofrim records), usually it is read from a printed text without a berakha.