Stochastic programming with decision dependent uncertainty is far less a beaten track than stochastic programming with fully exogenous uncertainty.
A short, not well thought through, answer to your question would be: No, you cannot apply Benders (L-shaped method) to such problems directly, at least not in all cases.
A hopefully more useful answer would be to study the specific problem at hand. Particularly, how does your decision $x$ influence $\xi$? Is your $\xi$ continuous or discrete? (probably the latter since you say scenarios).
There are at least two ways in which $x$ can influence $\xi$. It can influence the timing when $\xi$ is revealed (probably not your case), see for example this article.
Or it can influence the distribution function or support of your $\xi$ (probably your case, I guess). In the latter case, one can still distinguish a couple of cases:
- A. Where you can obtain infinitely many different distributions of $\xi$ as a result of $x$
- B. Where you can obtain finitely many distributions
For class B. see this article and, particularly, this article. Both are presented for a multistage setting, but can equally well describe your case (after all you have a multistage problem with only two stages). You will probably see that in case B. you might be able to use Benders though within a broader scheme (e.g., you might have to solve several stochastic programs). In case A. things are a bit more involved and I am not sure there is much out there.