Theorem 3.3 from p. 86 of "Cerone, Pietro, and Sever S. Dragomir. "A survey on bounds for the Gini Mean Difference." Advances in Inequalities from Probability Theory and Statistics (2008)" states that
$$R_G(f) \le \frac{2}{(q+1)^{1/q}}\left[M_{E,p}(f)\right]^{1/p}$$
where $R_G(f)=\frac{1}2 E|X-Y|$, $p>1$, $1/p+1/q=1$, and $M_{E,p}(f)=E\left[|X-\mu|^{p}\right]$.
The proof is short and uses Holder's inequality. Now, Remark 3.2 says to take $p=q=2$ in the inequality to find
$$R_G(f) \le \frac{2}{\sqrt{3}}\sigma$$
The reference says this inequality is known and refers to
https://galton.uchicago.edu/~wichura/stat304/handouts/L09.means3.pdf
But, I could not access that website. It also states the upper bound is obtained for the Unif(0,1) distribution. It seems like there is a misprint in the reference because I think the inequality should be $R_G(f) \le \frac{1}{\sqrt{3}}\sigma$. There is a $\frac{1}/2$ included as part of the definition of the Gini mean difference $R_G(f)$.

Theorem 3.3 from p. 86 of "Cerone, Pietro, and Sever S. Dragomir. "A survey on bounds for the Gini Mean Difference." Advances in Inequalities from Probability Theory and Statistics (2008)" states that
$$R_G(f) \le \frac{2}{(q+1)^{1/q}}\left[M_{E,p}(f)\right]^{1/p}$$
where $R_G(f)=\frac{1}2 E|X-Y|$, $p>1$, $1/p+1/q=1$, and $M_{E,p}(f)=E\left[|X-\mu|^{p}\right]$.
The proof is short and uses Holder's inequality. Now, Remark 3.2 says to take $p=q=2$ in the inequality to find
$$R_G(f) \le \frac{2}{\sqrt{3}}\sigma$$
The reference says this inequality is known and refers to
https://galton.uchicago.edu/~wichura/stat304/handouts/L09.means3.pdf
But, I could not access that website. It also states the upper bound is obtained for the Unif(0,1) distribution. It seems like there is a misprint in the reference because I think the inequality should be $R_G(f) \le \frac{1}{\sqrt{3}}\sigma$. There is a $\frac{1}2$ included as part of the definition of the Gini mean difference $R_G(f)$.