Can I clear chat using a command block?

In Minecraft 1.9, F3 + D will clear your chat.

Because this works, there must be either a command or .class that it calls upon. Can this somehow be run from a command block at a certain period of time?

It is not possible to clear the player's chat using commands.

However, you could dump a large amount of newlines (~100) using /tellraw, which will clear their current history but will retain the newlines.

/tellraw @a ["\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n"]