Raising money on the Kickstarter or GoFundMe websites
There have been a number of successfully crowdfunded research projects already, see e.g. this write-up. Specific crowdfunding websites have even been set up for research projects, such as Experiment and Petridish. As you might guess from all this activity there is no general issue with crowdfunding research. However, your particular university might have specific policies or restrictions to be aware of. (Your advisor should know the right people to get in touch with to find out for sure.) I've heard that a common roadblock is that the university's accounting offices might not know how to (or want to) handle this kind of projects, where the funding comes from many small donations, rather than a central funding agency. In any case, there are no obvious global restrictions that apply.

There have been some other crowdfunding questions on this website you might be interested in as well: What are some tips for crowdfunding science off websites like PetriDish?, and Can a PhD be crowdfunded?.