A definition of the dynamical system is that:
$\phi:R \times E \to E$ is a dynamical system where $\phi \in C^1$, $E$ open subset of $\mathbb{R}^n$, and if $\phi_t(x) = \phi(t,x)$, then $\phi_0(x) = x$ for all $x \in E$ and $\phi_t \circ \phi_s(x) = \phi_{t+s}(x) $ for all $s,t \in \mathbb{R}^n$ and $x \in E$ (Perko)
For instance, given: $$\dot x = Ax$$ $$x(0) = x_0$$
$\phi(t,x) = \exp(At)x_o$ defines a dynamical system on $\mathbb{R}^n$
Does the solution to an integro-differential equation of the form
$$\dot x = \int^t_0 f(x(s)) ds$$ $$x(0) = x_0$$
violate the definition of a dynamical system?
The answer is no; it does not violate the definition of a dynamical system.
You can define a typical dynamical system for the system $$\dot{x}=\int_0^t{f(x(s))ds}$$ by increasing the order of the state vector. Specifically, if we define the extra state $$\xi:=\int_0^t{f(x(s))ds}$$ then we have the augmented state vector $x_{ag}:=[\matrix{x^T & \xi^T}]^T$ with a the classical form $$\dot{x}_{ag}=f_{ag}(x_{ag}):=\left[\matrix{\xi \\f(x)}\right]$$ and initial condition $x_{ag}(0)=[\matrix{x_0 & 0}]^T$. Then it is standard to define the dynamical system for the new autonomous ODE system assuming locally Lipschitz property for $f$ by using Picard iteration.