'Backer' and 'supporter' are fine; 'champion, or even 'proponent' might work in a more formal context. But I suspect the question 'is there a verb for X?' may imply the underlying assumption 'there should be, because that's the way we do it in my language'...
In many cases I'd tend to say something like 'I'm on his side' rather than 'I'm his backer'.
b
PS You might be able to do something with the phrasal verb 'to egg on', but I've never heard of an 'egger on' (except maybe jokingly). When there are an organized and vocal group of supporters, we've borrowed the French claque, and a group of paid supporters who know nothing about what they're 'supporting' is routinely (especially in press reports) referred to as a 'Rentamob'. ;-) (This one is quite colloquial.)