Do colleges/professors sign contracts with educational publishers that require them to use the publisher's learning platform?
Is it possible? I suppose so, but that's not the way things normally work. Usually the professor, or a committee of professors, decides what textbook and/or homework system to use. There is not usually a contract, and they can change to a different book or system at any time.

I can think of several more likely reasons for using the system:


Simplicity. A generic online learning system like Blackboard has little pre-made content, while a system from a publisher will come with questions already written.  Writing even one quiz per week "from scratch" in Blackboard is highly time consuming. Giving quizzes on paper is also more time consuming, and takes class time. 
Textbook integration. The online systems from publishers are often tightly integrated with an online textbook. Students often complain about systems like Blackboard, saying they don't see the link between the questions there and the textbook.
A desire to try out online systems. From the question, it sounds like the professor could get more "value" from the system. Perhaps she's just trying it out, before really committing to it. Or perhaps she is concerned about complaints if she assigns more frequent assignments. Student feedback is important in this case. 
If a committee decided that the online system is required, perhaps the professor could be "playing along" by assigning a minimal amount of work on it, just enough to avoid criticism from colleagues.