Apache v2 s4 says that
You may reproduce and distribute copies of the
Work or Derivative Works thereof in any medium, with or without
modifications, and in Source or Object form, provided that You
meet the following conditions:
(a) You must give any other recipients of the Work or
Derivative Works a copy of this License
IANAL/IANYL, but giving them that licence locked up inside the firmware of a device with no written output capability is a pretty nugatory way of honouring it. If you take that route you had better pray that you don't accidentally make any other compliance slips, because your decision about the licence copy could be presented as evidence that you've decided to sail close to the wind in terms of compliance, ie, evidence of bad faith.
You say you're supplying no documentation; OK. You don't say where you're selling this device, but if you hope to sell it into the EU or US markets, and in my experience many others also, you will have to include a one-or-more-page set of regulatory statements (FCC compliance, that sort of thing). Adding the Apache licence on to that won't kill you, and it casts your licence compliance behaviour in a much better light.
Edit: you say the author of the software has approved your putting the licence in the online-available documentation, which is linked from a one-page sheet distributed with the product. If (s)he's the sole rights-holder, that is, if (s)he wrote the code in its entirety, then (s)he can permit you to use it under any terms (s)he wants, including "Apache with a reduced documentation requirement", which is apparently what (s)he's doing.
If (s)he based her code on someone else's Apache-licensed code, then (s)he can't free you from the obligations you owe to that earlier author, and my answer abve applies.