In fact, bacteria, which lack sexual characteristics, already do what you're describing. It's called horizontal gene transfer. Basically, living members of the population swap DNA with their neighbors. It's actually a pretty common activity, and helps mutations spread more quickly through asexually reproducing species.
That being said, typically smaller and simpler segments of DNA are swapped between members. True sexual reproduction transmits far more information, achieves a much greater amount of genetic diversity, and is more practical for organisms that don't reproduce every few hours.
The process could, potentially, work in macroorganisms, but complex organisms have literally billions more base pairs than the prokaryotes that commonly engage in this sharing. To share enough DNA to boost phenotypic diversity, DNA would have to be carefully packaged and transferred, likely in chromosome-like structures. Organisms would have to develop specialized external genetic transfer structures, and pretty soon you'd be back to sexual reproduction.