Rigid monoidal abelian categories are often called tannakian category; a tannakian category with a fiber functor (i.e., a strong monoidal exact functor) is referred to as a neutral tannakian category. I will adhere to this terminology.
It was proved by Grothendieck, Saavedra-Rivano and Deligne that the 2-category of tannakian categories over a fixed field $k$ is equivalent to the 2-category of affine gerbes over $k$. With each affine gerbe $\Gamma$ you associate the category of vector bundles on $\Gamma$. Conversely, given a tannakian category $\mathcal A$, the objects of the corresponding gerbe over a $k$-algebra $A$ are fiber functors from $\mathcal A$ to finitely generated projective $A$-modules.
So, if $\Gamma$ is an affine gerbe, fiber functors from the category of vector spaces on $\Gamma$ to vector spaces on $k$ correspond to objects of $\Gamma(k)$; hence, to give an example of a tannakian category without fiber functors it is enough to give an example of an affine gerbe without a rational point. There are many such examples; for example, you can take a surjective homomorphism of algebraic groups $E \to G$, and a $G$-torsor $P$ over $k$ that does not lift to an $E$-torsor. The gerbe of liftings of $P$ to an $E$ torsor is an affine gerbe, and does not have any rational points.