It is well-known that there are positive real numbers $\beta, \gamma$ such that
$$\vec{OA} + \beta \vec{OB} + \gamma \vec{OC} = \vec{0}.$$
So for positive integer $k$, we have
$$k \vec{OA} + k\beta \vec{OB} + k\gamma \vec{OC} = \vec{0}.$$
Let $m(k) = [k\beta]$, $n(k) = [k\gamma]$, where $[x]$ is the biggest integer which is less than or equal to $x$, and $\{x\} = x - [x]$.

Assume $T$ is an integer larger than $\max\left\{\frac{1}{\beta}, \frac{1}{\gamma}\right\}$. Then the sequences $\{m(kT) \mid k = 1, 2, \dots\}$ and $\{n(kT) \mid k = 1, 2, \dots\}$ are increasing, and
$$\begin{aligned} & |kT \vec{OA} + m(kT) \vec{OB} + n(kT) \vec{OC}| \\ & = |-\{kT\beta\} \vec{OB} - \{kT\gamma\} \vec{OC}| \\ & \leq |\vec{OB}| \cdot \{kT\beta\} + |\vec{OC}| \cdot \{kT\gamma\} \\ & \leq |\vec{OB}| + |\vec{OC}|. \end{aligned}$$
This shows there exists infinitely many vectors such that
$$kT \vec{OA} + m(kT) \vec{OB} + n(kT) \vec{OC},$$
whose endpoint is in a circle $O$ of radius $|\vec{OB}| + |\vec{OC}|$. So there are two of the vectors, such that the distance between the endpoints of the two vectors is less than $\frac{1}{2007}$, this means there exists two integers $k_1 < k_2$, such that
$$\begin{aligned} & | (k_2 T \vec{OA} + m(k_2 T) \vec{OB} + n(k_2 T) \vec{OC}) \\ & \quad - (k_1 T \vec{OA} + m(k_1 T) \vec{OB} + n(k_1 T) \vec{OC}) | \\ & < \frac{1}{2007}. \end{aligned}$$
So, if we let $p = (k_2 - k_1)T$, $q = m(k_2T) - m(k_1T)$, $r = n(k_2T) - n(k_1T)$, then $p, q, r$ are integers, and
$$|p \vec{OA} + q \vec{OB} + r \vec{OC}| < \frac{1}{2007}.$$