It's essential that the documents should be destroyed immediately.
Why can't ought to be used in place of should and why can't I go for would?
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The only one that fits grammatically is should.
Another idiomatic possibility is to insert no word at all - It is essential that the documents be destroyed immediately.
Ought to be, would and had better cannot be qualified by it is essential that...
If something ought to be done it ought to be done. It is not gradable and subject to essentiality qualifications. Similarly with had better.
If it would be done, it is outside the scope of anyone to alter anything - so essential/inessential does not apply.