My business contracts a payroll company to manage payroll and filing of 940/941 tax forms. Apparantly, the IRS notified me that several of these for FY 2016 were never filed. I contacted my payroll company who apologized for the screwup and "apparently" corrected this by recently filing them.
Do you have their apology in writing?
Generally, you can get penalties abated due to reasonable cause if the paid preparer admits they screwed it up in a statement you can provide to the IRS. That statement can be to you, via email or something similar, or directly from the preparer to the IRS. Despite the fact that the IRS rarely punishes preparers for making mistakes, most are extremely reticent to admit their errors, even directly to the IRS.
Here's the problem you face with 941s/940s:
If your payroll taxes are paid on time, then there is no penalty for late filing a 941 or 940. The fact that you hired a payroll company to prepare your payroll does not exempt you from your responsibility to pay your payroll taxes timely. You're supposed to notice if those payments didn't happen and immediately correct it.
Because of this, it can be very difficult to get the IRS to remove these types of penalties.
Some of the easier penalties to get removed due to reasonable cause is the failure to make proper tax deposits - either because you were unaware that you were a monthly payor or because you were unaware that you had an electronic payment requirement. In other words, if you made the payments, but didn't make them the correct way. Even then, you're screwed if you have a negative tax history or if you took a long time to fix it after receiving a notice from the IRS.
941/940 penalties are eligible for first time penalty relief - so you can get the earliest penalty removed without needing a cause - assuming a clean three year history prior to that.
You can't generally hold an accountant liable for the tax portion or the interest on the tax portion. But you can make a claim for penalties and the interest on the penalties. You could sue them, or request their E&O insurance info to make a claim.