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Choice B is correct. It’s given that a business owner plans to purchase $81$ chairs. If $p$ is the price per chair, the total price of purchasing $81$ chairs is $81p$. It’s also given that $7\%$ sales tax is included, which is equivalent to $81p$ multiplied by $1.07$, or $81\left(1.07\right)p$. Since the total budget is $\$\mathrm{14,000}$, the inequality representing the situation is given by $81\left(1.07\right)p\le \mathrm{14,000}$. Dividing both sides of this inequality by $81\left(1.07\right)$ and rounding the result to two decimal places gives  $p\le 161.53$. To not exceed the budget, the maximum possible price per chair is $\$161.53$.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect. This is the maximum possible price per chair including sales tax, not the maximum possible price per chair before sales tax.

Choice D is incorrect. This is the maximum possible price if the sales tax is added to the total budget, not the maximum possible price per chair before sales tax.