The literal answer to your question is probably yes.  Most state nonprofit corporation laws allow a nonprofit corporation to engage in any lawful activity, and, assuming that the gaming app does not involve illegal gambling, the company could be organized in a nonprofit form.

What I assume that you are really asking, however, is whether the company could qualify as a charitable organization exempt from federal income tax under Section 501(c)(3) of the Tax Code.  (See Ready Reference Page:  “What Do We Mean When We Say Nonprofit?”) To that question, the answer is probably no.  Charitable tax exemption is not based on how you use the income.  It is based on the activity that generates the income.  It is hard to see how producing a gaming app would be considered educational or charitable.  While the IRS accepts a broad range of purposes as charitable (See Ready Reference Page:  “IRS Has Generally Expansive View of ‘Chartable’”), I haven’t seen a citation that would suggest it would go this far.