Can the word "motivate"be used in a negative situation? For example, "the Stamp Act motivated people to voice their opposition"
The verb 'motivate' can simply mean to "influence a person or people to do something", with no "positive" or "negative" meaning:
to cause someone to behave in a particular way
However, it is often used to describe efforts to make people do more, do better, or make positive choices, and "motivational" speeches, posters, slogans, etc, are used for that kind of purpose:
to make someone want to do something well
"Motivation" is a neutral term. The same set of circumstances can be said to motivate someone to do either positive or negative acts.
Extreme poverty motivated her to start her own business.
Extreme poverty motivated her to commit several robberies.
In any case, "voicing opposition" is not normally considered a negative act. As with "motivate" it's a neutral expression whose connotations depend on the exact circumstances of the opposition.
For example, in the TV series "The Simpsons" the misanthropic owner of the town's nuclear power plant often voices his opposition to things like charity or quality of life, which most would consider to be a "negative" opinion:
Yes. Although motivation has a positive connotation, for instance when we talk about the reasons a murderer murdered someone, we talk about his motives. In a similar sense, politicians often advocate their policy as meant to diminish certain motivations, such as the motivation to misuse social benefits.