The board can fill the vacancy with someone who meets the eligibility requirements for the position according to your bylaws.
Notice is given to the body filling the vacancy (the board in this case) and could be included in the call of the meeting.
Ultimately it is up to your organization to interpret your rules. Generally when there is only one candidate and the bylaws do not require a ballot vote in that situation, the chair can simply declare the sole candidate elected.
In the 11th edition:
On page 618, line 34, the reference "p. 607, ll. 18-21" doesn't seem to be related to "the seating of delegates and alternates" from the previous line. Should it be some other reference?
How about this one from the In Brief book:
At the bottom of page 48 of RONRIB, the last word in the box is "resolution". Should it be "motion" instead?
Is there a citation for this? I can't find a clear citation of what to do when the body that elects is different than the body that can fill the vacancy.
See RONR 11th ed., p. 467, ll. 25-28.