You need to know that some rightsholder most probably owns the rights of every popular characters' copyrights and/or trademarks, and those rightsholders have the right to exclusively use and/or to license others to copy,make derivative works from, distribute, etc.those character rights, and to sue any unauthorized user for infringement, unfair competition, etc.
Lots of people on Avvo ask about making crafts out of someone else's IP, and lots of them seem to think that if a lot of other people are doing it, it must be ok, but it's not, and you may want to do a search and look at the other questions and answers.
But - there's almost always a but - tere are limits to these rightsholders' exclusive rights, including "parody" and "fair use," which require another user to "transform" the original work or comment on it in such a way that it's considered allowable. How much change is enough to transform a work? This can't be described or quantified because it always depends on the use at issue, and what's the difference between a transformative work and a derivative work? Generally, you can't try to trade on someone else's IP by copying it, you have to create something new and different enough to be recognized for it's own contributions. That means not just creating a painting of a character on a craft.
See an IP lawyer to fully disclose what you want to start out with and what you want to end up with to get some specific advice.
Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.
One can, but one risks being sued, or being threatened with a lawsuit and perhaps having to pay a settlement, for doing so. It is almost certainly a copyright infringement to do so.
It is frustrating to hear that answer, because you see this done all the time -- think of all the depictions of Calvin from Calvin and Hobbs that you see around -- yet the creator of that strip refuses to do any licensing at all. Yet the fact that people do it does not make it permissible, and it does not constitute a defense to the one who is "chosen" to get the attention of the rights holder.
All things considered, building a business on such a "model" is not a good idea.