Stormfront seems to think so.
Stormfront user Daedra posted a video from an apparent fan of the Iron Guard with the thread topic "Richard Spencer 2020." No, I'm not going to link to it.
But I will post screenshots.
You can find the YouTube video yourselves if you must. I don't recommend it. It's basically a mashup of several speeches by Richard Spencer cut with a collage of racial violence, celebrities, and nightclubs. It ends with text overlaying the eyes of a Roman statue that reads, "Revolt against the modern world" - a nod to Evola's book by the same title. All but the text fades to black and the text is joined by a black sun.
A few pages on, user Tenniel posts another video, purporting to be proof that Richard Spencer is going to run for president. The contents of the video are certainly less dramatic than the campaign video. In it, Spencer is asked by a listener to his podcast whether he is considering a run for president.
Spencer, in true form, makes the question about white nationalists running for political office in general. He sees a political run as potentially galvanizing for the white nationalist movement, probably not under the banner of the alt-right. The video ends with Spencer suggesting that he has been talking to James Edwards, founder of the anti-semitic Political Cesspool radio show.
Spencer is self-aware enough to recognize that the white nationalist movement is in retreat back to the webzines and think tanks it had previously occupied. However, he seems to believe that 3rd party candidates are accorded all sorts of rights. For starters, how would he get on the ballot anywhere? Who's signing that petition?
But I wish him well with it. It's yet another opportunity for him, and by donation the white nationalist movement as a whole, to squander money on another failed and likely violent spectacle. This time, they'll have to pay in bitcoin.