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.
Well, it could happen that the size of $\bigcup_{\xi\lneq\lambda^+}M_\xi$ is actually strictly below $2^{\aleph_0}$, (\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^+}$satisfying what you want. I once wrote something about elementary submodels which you can find at http://www.hausdorff-center.uni-bonn.de/people/geschke/publications/elsub.dvi and there is Alan Dow's [An introduction to applications of elementary submodels in topology, Topology Proceedings 13 (1988), pp. 17-72]. Also, the second volume of Discovering Modern Set Theory by Just und Weese (AMS Graduate Studies in Mathematics 18) has something about elementary submodels in set theory. 1 Well, it could happen that the size of$\bigcup_{\xi\lneq\lambda^+}M_\xi$is actually strictly below$2^{\aleph_0}\$, and in this case you cannot have what you want.