Absolutely impossible. There are several basic reasons:
1. Genetic modification isn't that simple
Not going to go into the mechanics of it, but suffice it to say that you can't perform genetic modification simply by having the cells of one species close to the cells of another. It requires complex techniques, such as shooting the new code at the host genome along with metal particles, shock or heat treating the receptor cell, etc.
2. This "feature" would not make a species more viable
If the species were around, just say for the past 1,000 years (+++), imagine the quantity of information that would have been gathered. Besides being unable to reproduce with others of its kind due to genetic incompatibility, the extra information being added every generation would make the species so laden with "features" that they would very quickly start to use their utility.
Furthermore, the gained "features" would tend to make the predator more like its prey. This would not help it and its progeny to survive; rather it would hinder them from catching additional prey. If a fox, for instance, begins to shrink because it has been eating mice, the time will quickly come when it will no longer be able to catch the mice.
3. Somatic cells are not reproductive cells
I'm aware this isn't necessarily in your question, and quite possibly you did not mean to imply it, but as long as I'm here, it's worth the effort to point out that in a lot of predatory organisms, reproduction takes place via meiosis, fertilization, etc. That is to say, even if the new DNA were to be absorbed by the cells and by some unbelievable chance, the code were to be copied into the genome of the host's cells, that trait would not be passed on to offspring unless the code were also implanted in the carrier's reproductive cells.
Even if that happened, you would once again have the issue that the additional codon sequences would very likely make it impossible for the reproductive cell to bind with another "equivalent" gamete.
Sorry, but no. Just - no.