I would not say that the suggested translation 'The orators exclaim in the forum.' is totally wrong, but it is definitely not idiomatic English. Something more like 'shout, shout out, cry out' would be much better. With an accusative object you could also say 'declaim' (although that's restricted to certain registers). Cf. the OLD entry s.v. exclāmō. (Generally I advise my students not to default to whatever word the Latin word has been borrowed as into English, since the English word will often have shifted the semantics a bit.)
It should not only be accepted, they should change it to be the preferred translation. In English we do not use "exclaim" as a standalone verb. Although dictionaries will say it's both transitive and intransitive, I doubt you will find an example of the latter that is not followed by a direct quotation of the words being exclaimed.
I think the purpose was to indicate this term survives directly, albeit archaically, into modern English; and to indicate that there is often more than one possibility.
Shout, etc., should also be allowed though.
I suspect the orators also simultaneously declaim and proclaim...
The English sentence doesn't look natural to me. What do English native speakers think? I'd use "speakers" instead of "orators" (accepting both possibilities) and try and find a more appropriate verb than the exact calque that looks really as a literal translation an English speaker would not use in this sentence.
My tone may sound harsh but it's not meant to offend anyone and certainly not the contributors. You made a great job and I understand a literal translation is part of Duolingo policies but, please, use when it's possible, words that are the modern English equivalents, keeping only latin words that are specific to Roman times (like forum, patrician, plebeian, sesterce and the like). My remarks have only one goal: help you to improve this beta-course to make it a great one, that will help teenagers and adults realise that Latin can be fun.