Yes, there are many functions with this property. In fact, these functions are dense in $C_0([0,1])$.

For every $g \in C_0([0,1])$ there is a unique $f \in C_0([0,1])$ such that $(-\Delta)^{\alpha/2} f(x) = g(x)$ for $x \in [0,1]$. This $f$ is given more-or-less explicitly: $$f(x) = \int_0^1 G_{(0,1)}(x, y) g(y) dy,$$ where $G_{(0,1)}(x,y)$ is the Green function for $(-\Delta)^{\alpha/2}$ in $(0, 1)$: $$
G_{(0,1)}(x,y) = \frac{|x - y|^{\alpha - 1}}{2^\alpha (\Gamma(\alpha/2))^2} \int_0^{R(x,y)} \frac{s^{\alpha/2 - 1}}{\sqrt{1 + s}} \, ds
$$
with $$ R(x,y) = \frac{4 x (1 - x) y (1 - y)}{|x - y|^2} \, .$$

*(Edited).*
In fact, the class of functions $f$ defined above constitute the domain of $(-\Delta)^{\alpha/2}$ on $(0, 1)$, with zero exterior condition, and therefore it is dense in $C_0([0,1])$. A rigorous proof of this claim is somewhat complicated, though.

The fractional Laplacian $-(-\Delta)^{\alpha/2}$ is the Feller generator of (the transition semigroup of) the symmetric $\alpha$-stable Lévy process $X_t$. Let $D = (0, 1)$ and let $X_t^D$ be the process $X_t$ killed when it first exits $D$. Since every boundary point of $D$ is regular, $X_t^D$ is a Feller process, and hence its Feller generator is a densely defined operator on $C_0(D)$. Denote this Feller generator by $L$.

By the general theory of strongly continuous semigroups, $f$ is in the domain of $L$ if and only if $f(x) = \int_D G_D(x, y) g(y) dy$ for some $g \in C_0(D)$, and in this case $g = -L f$. What remains to be proved is that if $f$ belongs to the domain of $L$, then $L f(x) = -(-\Delta)^{\alpha/2} f(x)$.

It was proved by Dynkin (in his brilliant book *Markov processes*) that $f \in C_0(D)$ is in the domain of $L$ if and only if the limit in
$$
\tilde{L} f(x) = \lim_{r \to 0^+} \frac{\mathbb{E}^x f(X(\tau_{B(x, r)})) - f(x)}{\mathbb{E}^x \tau_{B(x, r)}}
$$
exists for every $x \in D$ and it defines a function $\tilde{L} f$ in $C_0(D)$; in this case $\tilde{L} f(x) = L f(x)$ for all $x \in D$. (The operator $\tilde{L}$ is the *Dynkin characteristic operator*).

Now the important fact is that the definition of $\tilde{L} f(x)$ does not depend on $D$! (It does not matter whether we use the killed process $X_t^D$ or the free process $X_t$ in the definition, as long as $f = 0$ in the complement of $D$).

Furthermore, convergence in the definition of $\tilde{L} f(x)$ implies convergence to the same limit in the definition of $-(-\Delta)^{\alpha/2} f(x)$ as a singular integral (for any fixed point $x$). This phenomenon seems to be specific to the fractional Laplacian, and it appears that it has been first observed here.

Final remark: Regarding the converse claim: "if $f \in C_0(D)$, $(-\Delta)^{\alpha/2} f(x)$ is well-defined as a singular integral for each $x \in D$ and it defines a $C_0(D)$ function", I believe it is not stated explicitly in the literature unless $D$ is full space.