The UK requirements I am aware of (there may well be others) are:
The Electronic Commerce (EC Directive) Regulations 2002 Reg 6(1)(c):
"A person providing an information society service shall make available to the recipient of the service and any relevant enforcement authority, in a form and manner which is easily, directly and permanently accessible, the following information ... (c) the details of the service provider, including his electronic mail address, which make it possible to contact him rapidly and communicate with him in a direct and effective manner;"
NB there are a separate set of rules dealing especially with financial services (e.g. The Electronic Commerce Directive (Financial Services and Markets) Regulations 2002).
The Consumer Contracts (Information, Cancellation and Additional Charges) Regulations 2013 Reg 13(1)(a):
"Before the consumer is bound by a distance contract, the trader ... must give or make available to the consumer the information listed in Schedule 2 in a clear and comprehensible manner, and in a way appropriate to the means of distance communication used"
Schedule 2 includes:
"the geographical address at which the trader is established and, where available, the trader’s telephone number, fax number and e-mail address, to enable the consumer to contact the trader quickly and communicate efficiently;"
- There are other requirements to provide contact details (e.g. the GDPR, the Provision of Services Regulations 2009) but they are not so specific regarding the provision of email addresses.
- I have not investigated the extent to which these provisions apply to the particular type of company referenced.
- "Electronic mail", at least under some of the legislation, includes things like SMS messages.