But think about this: the deputy clerks are delegates of the county clerk's authority. To the extent of the delegation, the deputy clerks are, for all intensive porpoises, the county clerk.
Not constitutionally they aren't. And being as this is going to be a legal argument thing we'd have to look hard at the constitution. Obviously in practice this is exactly what they do, but it doesn't have a constitutional basis. That is apparent from the provision about a "vacancy in the office" ...
So, as long as a deputy of the county clerk is fulfilling the duties of the clerk, the clerk isn't "absent" for purposes of the statute, even when the person elected to that office is not, herself, present to fulfill the duties.
... so we'd need to consider whether the Clerk is "absent" independently of whether in practice the rest of the office is getting the day-to-day job done.
I guess the easiest way of doing that is if the judge/executive announces tomorrow morning that he is invoking 402.240 being as the Clerk is in jail. That would clarify everything immediately.
The second easiest way is for the judge/executive to in fact comply with 402.240 by memorialising the Clerk when she gets out of jail - that would defacto legitimise everything done in the meantime even there is some short-term uncertainty.
The hardest way is to do it all by legal challenges afterwards, which just leads to uncertainty all round - though I reckon 402.240 is strongly arguable right now.
Another consideration is 402.100(1)(a) and (c):
(1) A marriage license which provides for the entering of:
(a) An authorization statement of the county clerk issuing the license for any
person or religious society authorized to perform marriage ceremonies to unite
in marriage the persons named;
(b) Vital information for each party, including the full name, date of birth, place
of birth, race, condition (single, widowed, or divorced), number of previous
marriages, occupation, current residence, relationship to the other party, and
full names of parents; and
(c) The date and place the license is issued, and the signature of the county clerk
or deputy clerk issuing the license.
Notice that in (c) the signature may be that of the clerk
or a deputy - but in (a) the authorisation statement must be that of the clerk - a deputy will not do. That's the bit where the judge/executive needs to take over. But otherwise ...
402.105 Marriage license valid for thirty days.
A marriage license shall be valid for thirty (30) days, including the date it is issued, and after that time it shall be invalid.
... so once it is issued in accordance with 402.100 there is no problem with validity.
It is just that authorisation statement that is the problem, and that can be easily got over by invoking 402.240.