I dont know the specifics of what applies to airlines as I dont work for one but I'd note 3 things about your question:
1) VOR Tests, while included by pilots as part of the "required inspections" for IFR, are not really a "maintenance" issue or "inspection" like other IR inspections such as the pitot/static, elt, annual, etc so they aren't something the Engineering or Maintenance department is likely to be concerned with. They are generally conducted and signed off by the pilot flying the aircraft. In most cases this is an Instrument-rated pilot (or student) as the 30 day check procedure (including logging/signoff) isn't really discussed in private pilot curriculum but 91.171 itself does not really specify a requirement for who does this check. Unless it is radio repair station using a generated test signal (in which case they must be "certificated and rated appropriately" and must place an entry in the aircraft log in addition to the VOR check sheet), 91.171 does not list any real requirements of the person conducting the test. My reading of this is that as long as the test is conducted in accordance with the regulation, there is no reason your non-pilot friend sitting in the right seat cannot conduct and sign off on the VOR Test. Now I think most pilots, whether PIC or SIC, Captain or First Officer, wouldn't trust their navigation equipment and therefore life to just anyone (or even their First Officer/Co-pilot) and would check it themselves but it seems possible. Dual pilot-operations generally have both Captain and First Officer conduct the same tests as a cross-check and I again I doubt such dual operations groups would allow just anyone to conduct such a test. This is almost without a doubt true at an airliner.
2) The section that requires a 30 day VOR test is 91.171(a)2 but 91.171(a)1 lists no specific date requirement and states compliance is achieved when the VOR "is maintained, checked, and inspected under an approved procedure." 91.171(a)1 and 91.171(a)2 are "OR" regulations, meaning only 1 of the 2 needs to be complied with. I imagine this would cover most airliners since their planes undergo regular scheduled maintenance.
3) An aircraft as large as a 777 or 767 or A320 or really any "airliner" cannot be operated for hire under part 91, part 135 or even part 125 as it is too large to meet any of the definitions so even a "charter" airline flying for such a plane for hire would fall under the supplemental section of Part 121. I am unable to find any specific exclusionary grounds on why the aircraft couldn't operate part 91 as a private not-for-hire aircraft but I cant imagine there are many people out there flying them around in this manner. While the air carriers still ostensibly need to comply with part 91 regulations where undefined in part 121, there are many differences just within the regulations themselves (e.g. an operator under part 91 can initiate an approach to an airport when conditions are below minimums but a commercial operator under 135/125/121 cannot) but most importantly, the airline is free to deviate from Part 91 where necessary as long as it complies with their Ops Spec which must be filed with and approved by the FAA. Basically, the operate "outside of the law" and "write their own law" that is then signed off and sanctioned by the FAA as "compliance under the law" for that specific operation.
While I am unable to find anything specific about VOR testing under Part 121, there is a high likelihood that VOR testing is covered under the airlines Ops Spec allowing them to "skip" this check or is otherwise covered during regular maintenance by the "maintained, checked and inspected" clause of 91.171(a)1. Failing all of that it's an exceedingly easy and quick check to conduct and the pilots may still be conducting the test, even when compliant under other sections, using any of the prescribed methods. Many of the Class B and C airports capable of supporting such a large aircraft have VOR's right on the field and thus have good reception on the ground to conduct the dual VOR cross-check permitted by 91.171(c) at pretty much any time and location. Many also have designated VOR-system surface checkpoints which can be used under 91.171(b)2 or VOT systems which can be used under 91.171(b)1. The airlines also spend most of their time flying along specified routes where they probably regularly encounter designated airborne checkpoints to be used under 91.171(b)3 and/or can configure and use a known ground reference point under 91.171(b)4 (though exactly what an "acceptably low" altitude is and/or how to ensure you are directly over a ground fix when at 40,000 feet is unclear; I guess you could use another VOR station with DME in which case "directly over" should match your altitude).
Surprisingly enough, reading 91.171 I've found that one of the methods I was taught (and many other pilots I know in multiple regions of the country so not just my instructor's error) to conduct a VOR test, particularly useful in single VOR aircraft, is not in full compliance with 91.171. I was taught it was acceptable to plot a radial to known ground fix and we used it frequently at my previous airport where I did my instrument training and there was a prominent, off airway ground fix within 3 miles of the airport. The actual regulation of 91.171(b)4 read as a list of requirements not an "OR" and thus being on an established airway appears to be required. We also used the closest VOR station which was 10 miles to the North when the VOR station 20 miles to the South or even 35 miles to the East would appear to have been the FAA preferred station to use under 91.171(b)4(ii). I think low altitude and reception may have played a significant part in that though (It's been almost a year since I last flew there so dont really recall where and when we had sufficient reception to tune the VOR stations in question) and the regulation only states it as a preference so it would seem that it's not "unacceptable" so much as "less preferred."
Im also a little surprised (not really considering how long it takes the government to do anything) the Regs haven't been updated to allow a VOR cross check with a GPS receiver as this could be used just about anywhere provided sufficient GPS reception and would seem likely to be more accurate than any other airborne check and would not require the surveying and designation of ground checkpoints for the ground based checks.
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