Let us consider a vector field $X$ on a manifold $M$, here $M$ is without boundary, but not necessarily compact.
Citing well known literature, we know that $X$ has a flow, namely a function: $$ \gamma:M \times (-\epsilon,\epsilon) \to M $$
such that $\gamma$ solves the ordinary differential equation: $$ \dot{\gamma}_p(t) = X(\gamma_p(t)) $$ $$ \gamma_p(0) = p $$ This then supposedly generates a one parameter family of diffeomorphisms of $M$. In particular, for any fixed time $t \in (-\epsilon,\epsilon)$, we have the diffeomorphism defined by: $$ \gamma_t(p) = \gamma(p,t) $$ Roughly this means we take a solution to the ODE with initial condition $p$, push it along the solution trajectory until time $t$, and this new pushed-along point is what $p$ maps to under this diffeomorphism.
My question is as follows: It seems that to generate the one-parameter family of diffeomorphisms, we fix a point $p$, solve the ODE with initial condition $p$, and then move through time to get the new location of the point. How do we know there will be a uniform integral $(-\epsilon,\epsilon)$ such that this can be done? In particular, is there a statement (perhaps in the content of Picard's theorem) that says that a uniform time interval can be constructed that works for any choice of initial condition?
Your claim that any vector field admits a flow $M \times (-\epsilon, \epsilon) \to M$ is not correct. Actually, the existence of such a flow would imply, by stitching together integral curves, the existence of a global flow $M \times \mathbb{R} \to M$. A vector field admitting such a flow is called complete; however, not every vector field is complete. (Consider the vector field on $\mathbb{R}$ defined by $x \mapsto \frac{d}{dx} x^2$, where sufficiently large initial values blow up to infinity arbitrary quickly.)
If you look at an ODE textbook, you will find a statement that, assuming sufficient regularity on the vector field, guarantees for any point $p$ a neighborhood $V$ of $p$ around which the vector field admits a flow $V \times (-\epsilon, \epsilon) \to M$. (As a side-note: the fact that this flow is itself smooth is not simple to prove.) If the manifold is compact, then these neighborhood-wise guarantees can be collected to guarantee a flow $M \times (-\epsilon, \epsilon) \to M$, proving the important fact that every vector field on a compact manifold is complete.