How do you write Many people keep in touch with distant relatives by using the Internet.in imperative mood?

This sentence cannot be written in imperative mood. Imperative mood is used for commands or requests, including the giving of prohibition or permission, or any other kind of advice or exhortation.

An example of a verb in the imperative mood is be in the English sentence "Please be quiet".

Maybe you mean rewrite with a passive verb phrase. Then the sentence would be:

The internet is used to keep in touch with distant relatives.

You can add "by many people" if you want to.

The internet is used to keep in touch with distant relatives by many people.

Answer/

"Keep many people in touch with distant relatives, by using the Internet."