Show that if Automata $A$ recognize language $L$, there also exists an automata $A′$ that recognize $Double(L)$
You said you showed this result using regular expressions. Take the idea you used then and apply it to automata. You assume you're given an automaton $A$ for the language $L$. You need to construct an automaton $A'$ which accepts all and only the strings in $D(L)$.

Here's a hint to get you started: if you have a rule in $A$ of the form $\delta(p, a) = q$, meaning that in state $p$, seeing input $a$, you go to state $q$, how would you modify that rule so that from state $p$, seeing $a$ and $a$ again, will go to state $q$? Perhaps you might introduce another state along the way from $p$ to $q$?