Is it correct to write voted in in the following sentences?
- Members may vote in a new leader.
- Board members will be nominated and voted in by the team.
Both of those examples appear grammatically correct. The common usage of the compound "to vote in" is more frequently and naturally used in the form "to vote ... in." That said, ending a sentence in a preposition is to be avoided, so one usually follows "to vote ... in" with what position or office the object is being elected to.
In the case of your first example, that would be:
And in the case of your second example, because it is clear that board members are voted into the board, I would opt for the active voice instead of changing the format of "to vote in":
That said "to vote in" is most frequently used when one intends to specify what position the object is being voted into. If the electee is the main focus and the position is understood, "to elect" is more straightforward and natural.
In British usage, retaining the current wording I would have written:
although, in fact, I would probably have reworded it.
I agree with John Lawler's comment that voted in is best used after the election, as it means voted into office.
I would agree with the first and third suggestions by RedVillian, but would suggest
in place of his second suggestion; and, in place of his fourth suggestion,
In fact both the original suggestion, and RedVillian's second (and fourth?) suggestion are grammatically incorrect:
There is only one new leader, so only one Board member can be voted in.
Although the original could have meant (if 'the team' is different from the 'Board members') :