Well, it could happen that the size of $\bigcup_{\xi\lneq\lambda^+}M_\xi$ is actually strictly below $(\lambda^+)^{\aleph_0}$, and in this case you cannot have what you want. This happens for example if the continuum hypothesis fails, $\lambda=\aleph_0$, and all the $M_\xi$ are countable, a not so uncommon situation.
On the other hand, if you allow the $M_{\xi}$ to be sufficiently large, i.e., if you allow $\bigcup_{\xi\lneq\lambda^+}M_\xi$ to be of size at least $(\lambda^+)^{\aleph_0}$, then you can at least arrange sequences $(M_\xi)_{\xi\lneq\lambda^+}$ satisfying what you want.
I once wrote something about with the title "Applications of elementary submodels in general topology" which you can find at http://www.hausdorff-center.uni-bonn.de/people/geschke/publications/elsub.dvihttp://www.hausdorff-center.uni-bonn.de/people/geschke/publications and there is Alan Dow's [An introduction to applications of elementary submodels in topology, Topology Proceedings 13 (1988), pp. 17-72]. Also, the second volume of Discovering Modern Set Theory by Just und Weese (AMS Graduate Studies in Mathematics 18) has something about elementary submodels in set theory.
