In general, for the groups in a family of fixed Lie rank, there will be a bound on the degree of an alternating group that can occur as a subgroup, so the answer to your question is no. This is easily seen from the fact that they have representations of a fixed degree. For example $E_8(q)$ has a representation of degree $248$ over ${\mathbb F}_q$, so it cannot possibly contain $A_n$ for $n > 250$, and I would guess that there is a much lower bound than that.

Of course, if by a family you mean one of the doubly infinite families like $A_n(q)$ for arbitary $n$ then the acswer is yes, because, for each such family, by making $n$ sufficient large, the groups will contain alternating groups of arbitarily large degrees as subgroup of their Weyl groups.