Let $d\geq 0$ be a degree. Using the fact that the $L_1$ norm and the $L_{\infty}$ norm are equivalent on the finite-dimensional space of polynomials of degree $\leq d$, we see that there is a constant $c_d>0$ such that
$$\int_{[0,1]}f \geq c_df(0)$$
whenever $f\in{\mathbb R}[x]$ with degree $d$ and $f\geq 0$ on $[0,1]$.
I have computed the optimal value for $c_d$ for $d\leq 7$ :
$$ \begin{array}{|c|c|c|c|c|c|c|c|} \hline d & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline c_d & 1 & \frac{1}{2} & \frac{1}{6} & \frac{1}{8} & \frac{7}{90} & \frac{19}{288} & \frac{41}{840} & \frac{751}{17280} \\ \hline \end{array} $$
Is this sequence already known ? I found those values of $c_d$ by showing that $\int_{[0,1]}f$ can be written as a nonnegative linear combination of the $f(\frac{k}{d})$ for $0\leq k\leq d$, with the coefficient before $f(0)$ equal to $c_d$. This method breaks down for $d=8$ (and in fact, seems to break down for practically all $d\geq 8$) because negative coefficients appear in the decomposition.