Is there a grammatically correct sentence where all words are capitalized?

On Monday His Royal Highness Prince Edward, Duke of Kent, Googled World War Two ("Operation Overlord").

Sir, Yes Sir !! ... otherwise probably not. Mid-sentence verbs are rarely capitalized. EDIT - following your example, questions like Am I John Smith; or maybe a sentence full of letter abbreviations ... CPA, IT, MBA, DACA, COBRA (??)

I do not think so, mostly because you never capitalize verbs or conjunctions. The only case where this might fall through would be when a verb is at the start of the sentence.

This doesn’t seem to fit well with most modern sentences, and I think even if you could get one you would need another word for context that would not be capitalized.

For example a modern sentence might be “I said ‘No.’”

In the past maybe it would have worked to say “Said I, ‘Nay.’”

It seems like in modern speech a lot of verbs are no longer the first word said

“Running, he turned to me.” Here turned is the verb, not running

Even when the verb is the first part of the sentence it requires a word for context, and even then they do not sound like modern phrases.

“Swimming I am” am is required to state time subject (I) is swimming

“Swimming like a fish” this is a can of worms I cannot unpack

All this could be wrong, I have no credibility is the field of English, but my guess is that the longest sentence you are gonna get is one or two words longer than the proper nouns it contains.

That said, “I’m Lord Farquuad Regis Hempburn Yomsteed J. Samson” can get pretty long depending on titles and names.

There's nothing ungrammatical about using all caps. It can contravene style, like MLA or Chicago Manual of Style, but that's style, not grammar.