The information on the computer belongs to the Council, not the Clerk, so you should have arrangements in place to ensure that it can be made accessible without the involvement of the Clerk. This would usually be a sealed envelope containing all the relevant access codes etc, held by the Chair or another designated councillor.
For a prolonged period of sickness, the Clerk should assist in providing access, if there is a valid reason for this. For example, the council's email account(s) should be monitored at all times.
Did the Clerk offer an explanation for his or her reluctance to allow access?