Newly elected senators -- Republican or Democrat -- don't just get to pick their committee assignments. The two parties have somewhat different processes for assigning members to standing committees. In the GOP, it falls to the Republican leader (Mitch McConnell) and/or what's known as the "Committee on Committees" to decide who gets what.
"Current members on the committee will continue to be there. If there are other people that want to join, they can do so if there are vacancies," said Andy Fisher, spokesman for the Republicans on the Foreign Relations Committee.
The Congressional Research Service described the process in 2006. You can read the full report below, but the gist is fairly straightforward. Because of its power, prestige and popularity, Foreign Relations is one of four committees on which the GOP allows no members to overlap -- the others are Finance, Armed Services, and Appropriations. Republican members can only serve on one of the four. Beyond that, a seniority system prevails, with some newly elected members granted more seniority than others based on service in the House or as a governor. O'Donnell, of course, has neither.
The Republican leader does have some authority to appoint vacancies on desirable committees; however a spokesman for McConnell declined to comment on this story.
None of the eight current Republican members on the Foreign Relations Committee is retiring or in any danger of losing this November. If Republicans take a significant number of seats, the Democrat-to-Republican ratio of all the committees will change, and the Republicans will get to name one or two more members to Foreign Relations. Some existing members, and other newly-elected Republicans will no doubt express interest in those slots.
In other words, even if she wins, O'Donnell will face stiff competition.