Maintain a close working relationship with the president
Basically, you should be learning as much as possible about how to be the (eventual) president of your chapter.
Ask your president intelligent questions, learn how to plan your chapter’s annual events and logistics with those.
Begin correspondence with faculty advisor and school officials
Your president should include you on emails that they send regarding official business, and if possible, in meetings regarding chapter operations.
Develop ideas/initiatives/goals for your chapter
As president-elect, you have a unique perspective of not having many specific tasks assigned to you, but also being able to see how your chapter runs on a regular basis. Start to think about things that you would like to see your chapter develop and how you can implement them.
Perform any tasks assigned by the president
Advocacy - if your chapter doesn’t have a designated advocacy chair or committee, this is a great initiative for the president-elect to spearhead
Social media/PR - if your chapter doesn’t have these duties incorporated into a secretary/historian/public relations position, this is a great job for the president-elect
Correspondence between chapters - if your chapter plans or hosts events with other chapters/organizations outside your university, a good job for the president-elect could be to act as the correspondent to outside organizations