The inequality as stated is surely not true. Let $f(X_1, \ldots X_n) \in [n]$ be defined as the $j$ s.t. $X_j = 1$. Clearly, for a uniformly random permutation $f(X)$ is just a uniform number in $\{1, \ldots, n\}$, with expectation around $n/2$, but it is not concentrated: $\Pr(f(X) < n/4) \approx 1/4$; but in this example, clearly swapping two neighboring elements can change value of $f$ by at most $1$, so all $c_i = 1$, and you are asking for $f(X) = n/2 \pm O(\sqrt{n})$ with high probability.

On the other hand, using Azuma-Hoeffding inequality (which can also be used to deduce the McDiarmid inequality), we can get a similar in spirit statement, when instead we assume for all $k < \ell$, we have $$ |f(\pi) - f(\pi \circ (k, \ell))| \leq c_k. $$ Indeed, consider a random permutation $Y_1, \ldots Y_n$, each $X_i \in [n]$ (and they are also disjoint). For a function $f : S_n \to \mathbb{R}$ we can define a martingale, $X_i$ for $i \in \{0, \ldots n\}$, as $$ X_i = \mathbb{E}[f(Y_1, \ldots Y_n) | X_1, \ldots X_i], $$ s.t. $X_n = f(Y_1, \ldots Y_n)$ and $X_0 = \mathbb{E}[f(Y_1, \ldots, Y_n)]$.

To apply the Azuma inequality need to control martingale differences $$ |X_k - X_{k+1}| \leq c_{k+1}. \tag{1} $$ To do this, it is enough to show that for all valid $y_1, \ldots y_k$, and all $y_{k+1}, y_{k+1}'$ we have $$ |\mathbb{E}_Y[f(Y_1, \ldots Y_n) | Y_1 = y_1, \ldots Y_k = y_k, Y_{k+1} = y_{k+1}] \\- \mathbb{E}_Y[f(Y_1, \ldots Y_n) | Y_1 = y_1, \ldots Y_k = y_k, Y_{k+1} = y_{k+1}']| \leq c_{k+1}. \tag{2} $$ Indeed, we can find a bijection between all permuations that start with $(y_1, \ldots y_k, y_{k+1})$ and all permutations that start with $(y_1, \ldots y_k, y_{k+1}')$, s.t. each pair of the permutations differ by a single transposition of form $(k+1, \ell)$ for some $\ell > k+1$ (concretely $\ell$ will be a poisition of $y_{k+1}'$ in the first permutation). This bijection yields to a coupling between permutation $(Y_1, \ldots Y_n)$ conditioned on $Y_1 = y_1, \ldots Y_{k+1} = y_{k+1}$ and a permutation $Y_1', \ldots Y_n'$ conditioned on $Y_1' = y_1, \ldots Y_k' = y_k, Y_{k+1}' = y_{k+1}'$, s.t. with probability $1$ we have $$|f(Y) - f(Y')| \leq c_{k+1},$$ leading to a desired bound (2), and subsequently (1).

Applying Azuma inequality, in this case we can indeed deduce $$ \Pr(|f(X_1, \ldots X_n) - \mathbb{E}[f(X_1, \ldots X_n)]| > \lambda) \leq \exp\left(-\Omega\left(\frac{\lambda^2}{\sum_i c_i^2}\right)\right)$$

The inequality as stated is surely not true. Let $f(X_1, \ldots X_n) \in [n]$ be defined as the $j$ s.t. $X_j = 1$. Clearly, for a uniformly random permutation $f(X)$ is just a uniform number in $\{1, \ldots, n\}$, with expectation around $n/2$, but it is not concentrated: $\Pr(f(X) < n/4) \approx 1/4$; but in this example, clearly swapping two neighboring elements can change value of $f$ by at most $1$, so all $c_i = 1$, and you are asking for $f(X) = n/2 \pm O(\sqrt{n})$ with high probability.

On the other hand, using Azuma-Hoeffding inequality (which can also be used to deduce the McDiarmid inequality), we can get a similar in spirit statement. Namely

Lemma Let $f : S_n \to \mathbb{R}$ be a function s.t. for every permutation $\pi$ and every transposition $(k, \ell)$ with $\ell > k$, we have $$ |f(\pi) - f(\pi \circ (k, \ell))| \leq c_k. $$ Let $Y$ be a uniformly random permutation in $S_n$. Then $$ \Pr(|f(Y_1, \ldots Y_n) - \mathbb{E}[f(Y_1, \ldots Y_n)]| > \lambda) \leq \exp\left(-\Omega\left(\frac{\lambda^2}{\sum_i c_i^2}\right)\right). $$ Proof. Indeed, consider a random permutation $Y_1, \ldots Y_n$, (i.e. each $Y_i \in [n]$ and they are distinct). For a function $f : S_n \to \mathbb{R}$ we can define a martingale, $X_i$ for $i \in \{0, \ldots n\}$, as $$ X_i = \mathbb{E}[f(Y_1, \ldots Y_n) | Y_1, \ldots Y_i]. $$ In particular $X_n = f(Y_1, \ldots Y_n)$ and $X_0 = \mathbb{E}[f(Y_1, \ldots, Y_n)]$.

To apply the Azuma inequality need to control martingale differences $$ |X_k - X_{k+1}| \leq c_{k+1}. \tag{1} $$ To do this, it is enough to show that for all valid $y_1, \ldots y_k$, and all $y_{k+1}, y_{k+1}'$ we have $$ |\mathbb{E}_Y[f(Y_1, \ldots Y_n) | Y_1 = y_1, \ldots Y_k = y_k, Y_{k+1} = y_{k+1}] \\- \mathbb{E}_Y[f(Y_1, \ldots Y_n) | Y_1 = y_1, \ldots Y_k = y_k, Y_{k+1} = y_{k+1}']| \leq c_{k+1}. \tag{2} $$ Indeed, we can find a bijection between all permuations that start with $(y_1, \ldots y_k, y_{k+1})$ and all permutations that start with $(y_1, \ldots y_k, y_{k+1}')$, s.t. each pair of the permutations differ by a single transposition of form $(k+1, \ell)$ for some $\ell > k+1$ (concretely $\ell$ will be a poisition of $y_{k+1}'$ in the first permutation). This bijection yields to a coupling between permutation $(Y_1, \ldots Y_n)$ conditioned on $Y_1 = y_1, \ldots Y_{k+1} = y_{k+1}$ and a permutation $Y_1', \ldots Y_n'$ conditioned on $Y_1' = y_1, \ldots Y_k' = y_k, Y_{k+1}' = y_{k+1}'$, s.t. with probability $1$ we have $$|f(Y) - f(Y')| \leq c_{k+1},$$ leading to a desired bound (2), and subsequently (1).

Applying Azuma inequality, in this case we can indeed deduce $$ \Pr(|f(Y_1, \ldots Y_n) - \mathbb{E}[f(Y_1, \ldots Y_n)]| > \lambda) \leq \exp\left(-\Omega\left(\frac{\lambda^2}{\sum_i c_i^2}\right)\right),$$ as desired.