Normally operating agreements are kept in private files in a home or office. The law in Iowa legally requires the managers or manager-members of an LLC to maintain the records of a limited liability company in an organized fashion including an operating agreement, if any.
(In most countries in continental Europe, Latin America, Africa and Asia, these kinds of company records have to be kept instead in the custody of a notary public who has more record keeping responsibilities and education comparable to a lawyer, unlike a U.S. notary public. I don't know what the rule is in Puerto Rico and Louisiana, where the legal systems are based on Spanish and French law respectively, but have been influenced by the law of other U.S. states over the years.)
An operating agreement can't change the rights of the LLC vis-a-vis third parties. It is only binding on the members of the company, the company itself, and tax authorities. When you have only one member, the LLC is disregarded for tax purposes, so what the operating agreement says doesn't even affect the tax authorities. So, as a practical matter, there are almost never meaningful disputes over the authenticity or date of execution of an operating agreement in a single member LLC, and if there are, your signature and the date on the original operating agreement will be presumptively valid.
If you have more than one member, the presence of original signatures from more than one person on the operating agreement will typically be sufficient to authenticate the operating agreement in any dispute that arises between the members, between a member and the company, or with taxing authorities.
It doesn't hurt to have a copy somewhere as well, because an operating agreement is not like a dollar bill. The physical document isn't something with legal power, the physical document is only evidence of what you have decided to do and your decision is what has the actual legal power. So, as long as you can authenticate a photocopy of the original operating agreement that is just as good as the real thing, and normally identifying the fact that you signed the operating agreement will be sufficient to authenticate it.