Would a scheduled tweet or status update delivered by an automated process be considered a bot? If so then, yes, a bot has a right to free speech as it is an extension of the speech that a person wishes to express. The fact that it's not delivered, in person (so to say), by a human being doesn't abrogate the rights of that speech to be heard as that was the intent of the speaker in the first place.
Really, no different than the delayed reading of a printed newspaper editorial. The medium and method of delivery are irrelevant.
Of course, how "bot" is defined is what's pertinent here. Most of what people today would call "bots" are really just prerecorded messages where the automation isn't the speech itself but the means by which it is delivered.