With the Senate divided 50-50 and no vice president, how will it be organized?
Are there rules or precedents for this situation?
There will have been a president pro tempore elected prior to the loss of the vice-president as president of the Senate. That president pro tempore will continue until the Senate holds a new election.
A previous case of a tie, see below, resulted in an "arrangement" whereby an independent became president pro tempore.
Rules of the Senate
I. APPOINTMENT OF A SENATOR TO THE CHAIR
In the absence of the Vice President, the Senate shall choose a President pro tempore, who shall hold the office and execute the duties thereof during the pleasure of the Senate and until another is elected or his term of office as a Senator expires.
In the absence of the Vice President, and pending the election of a President pro tempore, the Acting President pro tempore or the Secretary of the Senate, or in his absence the Assistant Secretary, shall perform the duties of the Chair.
Secretary of the United States Senate
In certain parliamentary circumstances, the secretary may also preside over the Senate. The most recent occurrence was on 28 June 2010, after Senator Robert Byrd of West Virginia, who had been serving as President Pro Tempore died, and Vice President Joseph Biden was absent. On that occasion, Secretary of the Senate Nancy Erickson took the chair briefly until the Senate adopted a resolution to elect Senator Daniel Inouye of Hawaii as the new president pro tempore.
The President Pro Tempore of the Senate: History and Authority of the Office. September 16, 2015
Election to the Office
The usual practice of the Senate has been to elect as its President pro tempore a candidate of the majority party—almost invariably by a party-line vote or, in recent years, without a record vote. Most often, that person has continued to serve in the post so long as his party remains in the majority.
On a few occasions, the majority party has experienced difficulty in electing its candidate. Late in 1881, for example, Democrats in the Senate refused to permit administration of the oath of office to several Republicans waiting to be sworn in as Senators. As a result of this, the Democrats maintained a narrow majority of the votes in the chamber and proceeded to elect one of their own, Delaware’s Thomas F. Bayard, as President pro tempore. Even after the missing Republicans had been installed, the Senate remained equally divided between the two major parties. An arrangement was eventually agreed upon and an independent Senator, David Davis of Illinois, was elected to replace Bayard as President pro tempore.