There are two distinct issues here, and it should be possible to avoid being too prescriptive.
One is the question of communications from the clerk to councillors, especially the routine delivery of agendas and backing papers. Ten years ago, it would have been normal to deliver everything as hard copy. Now, most councillors use email, and are agreeable to receiving documents electronically. This saves a lot of time and expense for the clerk. No councillor should be forced to use email, though. There is no legal basis for such an enforcement, and it is unreasonable to expect people who genuinely feel uncomfortable with email or simply do not have the relevant facilities to accept email. If councillors find it acceptable, it's a good idea to make backing documents available on the council web site, rather than hard copy. Assuming the agenda is also available, that keeps citizens fully informed as well as councillors. For this purpose, I cannot see any reason not to use councillors' own personal email accounts, since the information flow is entirely from the council to the councillor, and consists of nothing but public documents.
The other is provision of an email address that is to be made public. This is not obligatory. It may be perfectly adequate to give phone numbers and addresses so that people can contact councillors. If emails are wanted, then it is probably best for them to be provided by the council. The addresses for councillors that do not wish to use them can be forwarded to the clerk (although that may generate some additional work, the volume is probably not large). There is a technical IT point that it is not actually necessary to insist on webmail. Provided the mail boxes are run using an IMAP server, all the mail is held on the server until deleted. Anyone requiring access to the mail can do so by looking at the server, and has no need to look at individual computers. Most mail is handled this way nowadays, and smartphones and tablets have mail apps that will work with IMAP mail servers, as will any computer. (The alternative is POP3, which is much less used than formerly).