What a terrible business model!

By the Chicken McNugget Theorem We know that the maximum number of pens that cannot be bought in this fashion is $6 \times 7 - 6 - 7 = 29$ . Thus $N + 5 \le 29 \Longrightarrow N \le 24$ .

Now $24 = 4 \times 6, 25 = 3 \times 6 + 1 \times 7, 26 = 2 \times 6 + 2 \times 7, 27 = 1 \times 6 + 3 \times 7$ and $28 = 4 \times 7$ , so with $N = 24$ each of $N, N + 1, N + 2, N + 3$ and $N + 4$ pens can be purchased, but not $N + 5 = 29$ . So $N = 24$ is a solution to this problem, but we must also show that it is the unique solution.

We are thus looking for a sequence of $5$ consecutive positive integers each of which can be represented as $6a + 7b$ for some non-negative integers $a,b$ but with the next greatest consecutive integer unable to be represented in this way. Now the integers $\le 29$ that can be represented in this way are

$6, 7, 12, 13, 14, 18, 19, 20, 21, 24, 25, 26, 27, 28$ .

The only subsequence of $5$ consecutive integers in this sequence is $24, 25, 26, 27, 28$ , which confirms that $N = \boxed{24}$ is the unique solution.