That's an interesting question. I'm just a native speaker, not a grammarian. Part of me says something is either normal or it isn't. Either it's normal or it's abnormal. It's a categorization. That would make it "ungradable." That's probably your answer.
But I think it falls in a grey area. All categories have imprecise boundaries. Some quick text searches confirm what I thought: the specific phrase "very normal" is common.
A celebrity named Kelly Ripa said "We just had a very normal, very regular wedding," I
n an 1800s novel the respected British author Hugh Walpole writes "He was a very normal, well-balanced young man and thoughts of this kind were unlike him."
I think it is only the specific phrase "very normal" that is common.