Let $P_d$ be the set of all probability measures $\mu$ on $\mathbb R$ whose first $d$ moments $c_1,\dots, c_d$ are the same as those of the standard normal distribution $\gamma$. By Theorem 3.1 in [1] (and the sentence following it there), there is an extreme measure $\nu$ of $P_d$. By Theorem 2.1 and Example 2.1 (a) in [1] or Corollary 11 in [2], $\nu$ is a mixture of finitely many Dirac measures. So, the support set of $\nu$ is bounded and hence contained in the interval $[-b,b]$ for some real $b>0$. That is, the point $(c_1,\dots, c_d)$ is in the convex hull of the set $\{(t,t^2,\dots,t^d)\colon t\in[-b,b]\}$.

This provides a positive answer to the question if the $[0,b]$ is replaced by $[-b,b]$. The answer to the original question is an obvious no, since for any random variable $X$ with support set in $[0,b]$ and $EX^2=\int_{\mathbb R}x^2\nu(dx)=1>0$, one would have $EX>0\ne\int_{\mathbb R}x\nu(dx)$.

Let $P_d$ be the set of all probability measures $\mu$ on $\mathbb R$ whose first $d$ moments $c_1,\dots, c_d$ are the same as those of the standard normal distribution $\gamma$. By Theorem 3.1 in [1] (and the sentence following it there), there is an extreme measure $\nu$ of $P_d$. By Theorem 2.1 and Example 2.1 (a) in [1] or Corollary 11 in [2], $\nu$ is a mixture of finitely many Dirac measures. So, the support set of $\nu$ is bounded and hence contained in the interval $[-b,b]$ for some real $b>0$. That is, the point $(c_1,\dots, c_d)$ is in the convex hull of the set $\{(t,t^2,\dots,t^d)\colon t\in[-b,b]\}$.

This provides a positive answer to the question if the $[0,b]$ is replaced by $[-b,b]$. Actually, the answer will remain positive for any probability measure on $\mathbb R$ in place of the standard normal distribution $\gamma$.

The answer to the original question is an obvious no, since for any random variable $X$ with support set contained in $[0,b]$ and $EX^2=\int_{\mathbb R}x^2\nu(dx)=1>0$, one would have $EX>0=\int_{\mathbb R}x\nu(dx)$.

Let $P_d$ be the set of all probability measures $\mu$ on $\mathbb R$ whose first $d$ moments $c_1,\dots, c_d$ are the same as those of the standard normal distribution $\gamma$. By Theorem 3.1 in [1] (and the sentence following it there), there is an extreme measure $\nu$ of $P_d$. By Theorem 2.1 and Example 2.1 (a) in [1] or Corollary 11 in [2], $\nu$ is a mixture of finitely many Dirac measures. So, the support set of $\nu$ is bounded and hence contained in the interval $[-b,b]$ for some real $b>0$. That is, the point $(c_1,\dots, c_d)$ is in the convex hull of the set $\{(t,t^2,\dots,t^d)\colon t\in[-b,b]\}$.

Let $P_d$ be the set of all probability measures $\mu$ on $\mathbb R$ whose first $d$ moments $c_1,\dots, c_d$ are the same as those of the standard normal distribution $\gamma$. By Theorem 3.1 in [1] (and the sentence following it there), there is an extreme measure $\nu$ of $P_d$. By Theorem 2.1 and Example 2.1 (a) in [1] or Corollary 11 in [2], $\nu$ is a mixture of finitely many Dirac measures. So, the support set of $\nu$ is bounded and hence contained in the interval $[-b,b]$ for some real $b>0$. That is, the point $(c_1,\dots, c_d)$ is in the convex hull of the set $\{(t,t^2,\dots,t^d)\colon t\in[-b,b]\}$.

This provides a positive answer to the question if the $[0,b]$ is replaced by $[-b,b]$. The answer to the original question is an obvious no, since for any random variable $X$ with support set in $[0,b]$ and $EX^2=\int_{\mathbb R}x^2\nu(dx)=1>0$, one would have $EX>0\ne\int_{\mathbb R}x\nu(dx)$.