A high-speed communication equipment typically would not be needed for

High-speed communication equipment is typically needed where large bandwidth, fast transfer rates, or fast local/network communication are important.

A. E-mail: E-mail is generally a low-bandwidth, asynchronous service. Text mails and ordinary attachments do not normally require high-speed communication equipment.
B. Transferring large volume of data: Large data transfer needs high bandwidth and faster data rates, so high-speed equipment is useful or necessary.
C. Supporting communication between nodes in a LAN: LAN communication is built around network equipment such as switches/NICs and standard fast local links. So this is not the best example of a case where high-speed equipment would not be needed.
D. More than one of the above: Only e-mail clearly fits the phrase "would not be needed". Therefore this option is incorrect.
E. None of the above: This is incorrect because e-mail is a valid case.

Therefore, the correct answer is Option A: e-mail.