From time to time, I cannot access a reference of an article that I am reviewing, either because my university did not subscribe to some journals (and I failed to find it myself on sci-hub, Google, and other places), or because the article doesn't seem to be available online.
Pinging the authors of the article (either the one that I am reviewing or the one that was cited) may work, but not always: legal issues (some publishers do not allow sharing papers online), unanswered emails, blind review preventing from contacting authors, etc.
Shall I just ignore the cited article in that case, even though it could be useful to review the article? Or are there other options, aside from asking my library to pay for it (hoping they would agree)?
While online access is very useful, it is not the only access. Every university I have been affiliated has provided an ILL service either at no cost or a nominal charge (e.g., a $1 processing charge). US University libraries can make requests to both the Library of Congress and the National Library of Medicine. I am less familiar with the ILL system in the UK/Britain, but I believe ILL requests can be made, possibly through a university, to the British Library.
There are of course some articles/books that are truly hard to find (e.g., not held by the LoC with deceased authors). In these cases, it seems reasonable to request a copy through the handling editor. You should never contact the authors of a manuscript under review directly.