I'm a competent native speaker, and having found the source of this text (a SATS test), I'm not sure even I can be sure which answer is "best"...
The committee requested that there be input from all the staff before taking a vote.
A ...from all the staff before taking a vote.
B ...from all the staff taking a vote.
C ...before the staff took a vote on the input.
D ...from all the staff before having taken a vote.
E ...from all the staff who would take a vote.
(The implication being you could select A if you think the original is "better" than any alternatives.)
Personally, I would immediately dismiss D as ungrammatical, and C as far too stylistically ungainly. And I don't like B or E because they both simultaneously use staff to reference...
individuals - who can provide input
a collective group - who can take a vote
It's possible not all native speakers object to this "zeugma". But all would agree that individuals cast votes, and groups (the electorate) take votes - never the other way around.
TL;DR: Yes, the original is in principle ambiguous. But pragmatically that's not really a problem, since if the vote in question was a staff vote, one would expect more elegant phrasing to avoid the awkwardness of using staff in two different senses...
The committee requested that there be input from each staff member before taking a [staff] vote.