Let $X(t)$ denote the number of occurrences of the event "a dot matrix printer needs a new ribbon" in the interval of time $[0, t]$. The ribbons are replaced at a rate of $10/7$ ribbons/week and it follows that $X(t)$ ~ $POI(\lambda t)$ with $\lambda = 10/7$. Also, $X(t)$ ~ $P_n(t) = \frac{e^{-\lambda t}(\lambda t)^n}{n!}$ where n is the number of occurences of the event "a dot matrix printer needs a new ribbon". Therefore, the probability of exhausting a stock of $5$ ribbons where $t=1$ week should be the probability that $5 \le n \le 10$ which can be calculated as follows $$\sum_{n=5}^{10} \frac{e^{-\frac{10}{7}}(\frac{10}{7})^n}{n!} \approx 0.0154$$
However, my textbook gives an answer of $.008$. Where did I go wrong?
You modelled the event differently from your textbook.
You modelled the event "printer A needs a ribbon" as an event that occurs with a known constant rate $\lambda$ and independently of the time since the last event. According to your model, there is some chance that printer A will need two ribbons in that week.
Your textbook assumed that printer A needs a ribbon every seven weeks. This is not an average. You know that if you change the ribbon of printer A today, you will need to change it again in seven weeks, not more, not less. What you don't know is when the ribbon was changed for the last time.
The result the book shows is, with $p=1/7$:
$$\sum_{n=5}^{10} \begin{pmatrix}10 \\ n\end{pmatrix} p^n (1-p)^{10-n} \approx 0.008$$