On the Fourier-Laplace transform of compactly supported distributions

Besides the direct approach to such continuity questions as in Nate Eldredge's comment one can try to use a closed graph argument. The closed graph theorem holds for ultrabornological domain spaces (like the space $\mathcal E'(\mathbb R)$ of compactly supported distributions endowed with the topology of uniform convergence on bounded subsets of $\mathcal E(\mathbb R)$) and webbed range spaces -- a very large class of locally convex spaces containing all Frechet spaces and thus, in particular, the space of entire functions. By Nate's remark that pointwise convergence is for free the Fourier transform $f\mapsto \hat f$ has closed graph and is thus continuous.

Of course, this is locally convex overkill -- but avoids concrete calculations.

Let me expand Nate Eldredge's comments. A more elementary proof than the one provided by Jochen Wengenroth, still requiring no computations, is as follows:

$\mathcal{E}'$ is the dual space of $C^\infty$, which is endowed with the Fréchet space topology given e.g. by the seminorms $\|f\|_k:=\|f\|_{C^k(B_k(0))}$. The fact that $f_n\to f$ in $\mathcal{E}'$ means that $\langle f_n,\phi\rangle\to\langle f,\phi\rangle$ for any $\phi\in C^\infty$.

Now the uniform boundedness principle holds also for linear maps from a Fréchet space to (say) a normed vector space. The proof is the same as that for the Banach space version; for details, see Theorem 2.6 in Rudin's Functional Analysis (2nd edition). Hence the $f_n$'s are equicontinuous, i.e. there exists $k\in\mathbb{N}$ and $C>0$ such that $$ |\langle f_n,\phi\rangle|\le C\|\phi\|_{C^k(B_k(0))}. $$ This gives $|\widehat{f_n}(\xi)|\le C\|e^{-2\pi i\langle\xi,\cdot\rangle}\|_{C^k(B_k(0))}\le C'(1+|\xi|^k)e^{2\pi k|\xi|}$ for $\xi\in\mathbb{C}$. Also, $\widehat{f_n}(\xi)\to\widehat{f}(\xi)$ for every $\xi\in\mathbb{C}$ (since it corresponds to evaluation of $f_n$ on the test function $e^{-2\pi i\langle\xi,\cdot\rangle}$). So you have a locally equibounded sequence of pointwise converging holomorphic functions and this implies your claim. This clearly works in any dimension.