Is it possible to construct a finite state automata for a decimal adder?
Interestingly, the question in the body has a negative answer, but the question of the title has a positive answer, if you choose the appropriate representation for the data.
Let me explain this for a binary adder (I let you generalize the argument for a decimal adder). First represent the numbers $x$ and $y$ to be added in reverse binary, with a final $0$ and make sure they have the same length by adding further $0$'s at the end if needed.
For instance, $22 = 2 + 4 + 16$ would be represented by $011010$ and $13 = 1+4+8$ by $101100$. Their sum $35 = 1 + 2 + 32$ is represented by $110001$. Now, just read this data column by column
\begin{matrix}
22 \to &amp;0&amp;1&amp;1&amp;0&amp;1&amp;0\\
13 \to &amp;1&amp;0&amp;1&amp;1&amp;0&amp;0\\
35 \to &amp;1&amp;1&amp;0&amp;0&amp;0&amp;1
\end{matrix}
to get
$(0,1,1)(1,0,1)(1,1,0)(0,1,0)(1,0,0)(0,0,1)$. Taking all representations of the triples $x, y, z$ such that $x + y = z$, you get a regular language on the alphabet $\{0,1\}^3$, recognised by the following automaton

The trick behind this representation is that addition can be obtained by a sequential transducer.