What are the legal Dispatching Requirement/s as far as Boeing is concerned for the aircraft to be used by the Crew (in this case on the 737) for example to perform a DUAL CHANNEL A/P (ILS) Approach right through the Flare and Roll-out?
Is a Placard in the Cockpit required stating CAT II or CAT III APPROVED?
Is a specific Techlog Entry/Release or in the Aircraft Cockpit Manuals a specific reference required?
Boeing does not make the legal requirements; your regulatory agency does. How your airline implements those regulations is set out in your Flight Operations Manual and Aircraft Operating Manual (or whatever they are called in your airline).
The requirements for a Cat I or II or III Autoland will be in those documents. You may have to look at both of them as well as the DDG/MEL to get the complete answer.
In my airline the cockpit placard is controlling for Cat II or III approaches. The Aircraft Log and DDG (along with any handy reference charts that may be in the other manuals) will tell whether a Cat I Autoland is allowed.