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@johnicahorton wrote:

... I have looked at the form instructions and they seem a little vague in that they do not state that they must be paid by year end, filing time, or January 15...? (since this is the final quarterly estimated due date for the prior year...?)...

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There is no ambiguity in these rules.  If they seem vague to you, you should read them a couple more times (and use materials from other reliable sources) until you understand them.  The responses above should, hopefully, set you on a more solid footing.

You must also remember that each quarter's ES-tax payment must be made on a timely basis.  Late payment for each quarter will continue to accrue penalties (cumulatively) until that underpayment is fully settled.  In other words, you could have a client who is due a refund subject to underpayment of ES-tax penalty because the required quarterly ES-tax payments were made after the respective due dates.

There is also an exception for the 4th quarter's ES-tax payment; that quarter's payment does not need to be made if the balance due is paid with a return to be filed by Jan 31.

Finally, ES-tax penalty may sometimes be mitigated by annualization.  This is particularly useful if you have clients who do not receive their income ratably over the year.  This could be because they received an extraordinarily large amount of income from bonus, stock options, trading, etc. in certain quarter(s).  The computation is relatively complicated and this is where you would charge your client for the cost of running those calculations.