The ability to autoland and the ILS Category are two separate issues.
An aircraft can be certified to autoland (within the certified wind and glideslope angle limits) after an auto-coupled ILS approach, regardless of the standard (CAT I, II or III) the ILS ground equipment is certified to. If a certain ILS system cannot be used for autolands it must be specifically stated in the approach plate for the runway involved.
An aircraft operator's manual and/or operations manual will state down to which visibility and to which DH autolands may be executed from a CAT I, II or III ILS.
Mind you: an ILS can be certified to CAT III standard at all times, only if all other requirements for CAT III operations are satisfied (safeguarding of the ILS critical area etc.) can CAT III ILS approaches to the associated minima actually be carried out.
To answer you initial question: there is no requirement to be CAT III qualified (aircraft, crew and ground equipment) in order to execute an autoland from a CAT I or II approach.