"that" introduces a "that" clause; its purpose is basically to allow a clause (here "your paper invoices and receipts are no longer sent to you in post") to work like a noun, for example so it can be the subject or object of a verb, in this case the object of "notice".
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JigneshbharatiOver the next few months you will notice that your paper invoices and receipts are no longer sent to you in post.You should replace "in" with by, before "post".