Here are explicit equations in characteristic not $2$. If the elliptic curve $E$ has equation $y^2=p(x)$ where $p(x)$ is a degree-$3$ polynomial, then let $r_1,r_2,r_3$ be the roots of $p(x)$. For any point $P=(P_x,P_y)$ on $E$ with $P_x\notin\{r_1,r_2\}$, let $s_1$ and $s_2$ be any square roots of $P_x-r_1$ and $P_x-r_2$, respectively, and put $s_3:=-P_y/(s_1s_2)$. Write $u:=s_1+s_2+s_3$ and $v:=s_1s_2+s_2s_3+s_3s_1$. Then $R:=(P_x+v,-P_y-uv)$ is a point on $E$ such that $2R=P$.

Note that there are known formulas for the roots of a degree-$3$ polynomial, in terms of square roots and cube roots of certain elements. Also, the hypothesis $P_x\notin\{r_1,r_2\}$ can always be achieved by relabeling the $r_i$'s if needed. So this recipe really does yield explicit equations.

Here are explicit equations in characteristic not $2$, from the paper "Division by 2 of rational points on elliptic curves" by Bekker and Zarhin (St. Petersburg Math. J. 29 (2018), 683-713).

If the elliptic curve $E$ has equation $y^2=p(x)$ where $p(x)$ is a degree-$3$ polynomial, then let $r_1,r_2,r_3$ be the roots of $p(x)$. For any point $P=(P_x,P_y)$ on $E$ with $P_x\notin\{r_1,r_2\}$, let $s_1$ and $s_2$ be any square roots of $P_x-r_1$ and $P_x-r_2$, respectively, and put $s_3:=-P_y/(s_1s_2)$. Write $u:=s_1+s_2+s_3$ and $v:=s_1s_2+s_2s_3+s_3s_1$. Then $R:=(P_x+v,-P_y-uv)$ is a point on $E$ such that $2R=P$.

Note that there are known formulas for the roots of a degree-$3$ polynomial, in terms of square roots and cube roots of certain elements. Also, the hypothesis $P_x\notin\{r_1,r_2\}$ can always be achieved by relabeling the $r_i$'s if needed. So this recipe really does yield explicit equations.