Unless the President publicly says something like ‘organization x supports this candidate!’. But even that would likely just cause a fuss where other Board Members of the organization would say that this wasn’t true, the organization had taken no official action to support a candidate. Then it becomes an internal argument for the organization.
I don’t even agree with your contention that this is somehow a “conflict of interest”. Serving as President of a non-profit organization certainly does not restrict your individual free-speech rights to support a candidate, speak out for that candidate, or even to take a paid position on their campaign staff. (The only conflict I could see is if the non-profit President is a paid position, and the employment contract prohibits any other outside employment.)
As long as the CEO makes it reasonably clear that the organization itself is not supporting the candidate, and the organization is not supporting the CEO’s work for the candidate, it should be OK, and this kind of thing happens fairly often. The first part can be tricky if the CEO and organization are seen as nearly identical, in which case the CEO needs to spend some time making the difference clear in any public communication. The second part can be tricky if the CEO is getting paid full-time by the non-profit but spending a lot of time on the campaign; a careful non-profit would make it clear how much time the CEO should be spending on the non-profit’s work, and grant some amount of unpaid leave to the CEO to do political work. Obviously the CEO shouldn’t be using non-profit resources on political work.
Whether it’s a conflict of interest is up to the board to decide. There are a lot of issues that we can’t opine about without knowing what the 501c3 does. And the board need to decides if the CEO can do both jobs at the same time.
But as t-bonham says, there’s a difference between him working for a candidate as an individual and creating an alliance between the organization and the political activity. I don’t think the conflict is inherent, but it’s something that should be considered.
As to factual question raised by the OP: is it possible to do both? It’s not prohibited outright. If you have concerns, raise them.
Just reiterating: the time diverted away could be a legal issue; if the non-profit is paying a full-time salary to the CEO, the CEO needs to be working full-time for the non-profit. The CEO can, in their time when they’re not working for the non-profit, do whatever they want, but they can’t work for the candidate during time that they’re paid by the non-profit. Otherwise, every campaign staff and lobbyist would be getting paid by a 501(c)3 while ‘volunteering’ for (and actually spending all their time on)the political work.
So the board needs to have a clear understanding with the CEO of how much time will be spent on the non-profit and on political work, and whether the current salary is appropriate given that split.
Of course the questions about effects on donors, the general reputation of the non-profit, and whether the board wants the CEO to be distracted during a tough time for the non-profit are all important ones, too.