Let $f(x)=2x^3+3x+k.$ We want the equation $2x^3+3x+k=0$ to have two distinct real roots in $[0,1]$.

Check the monotonicity of $f(x)$. Differentiate: $f'(x)=6x^2+3=3(2x^2+1).$ Since $6x^2+3>0 \quad \text{for all real } x,$ $f(x)$ is strictly increasing on all of $\mathbb{R}$.

A strictly increasing function can intersect the $x$-axis at most one point. That means the equation $2x^3+3x+k=0$ cannot have two distinct real roots anywhere on the real line, and hence certainly not in $[0,1]$.

Therefore, there is no real number $k$ for which the equation has two distinct real roots in $[0,1]$.

Hence the correct option is: $\boxed{\text{D: does not exist}}$