The Bristol Post is wrong - punctuation is part of grammar. The declarative sentence "The cat sat on the mat?" is ungrammatical because it's mispunctuated.
To be fair, it wasn't the paper saying it. They were just quoting North Somerset performance poet Melanie Branton.
In other news, "The cat sat on the mat?" isn't ungrammatical at all. It just isn't a declarative sentence. Imagine if someone had a cat that would happily sit on carpets, chairs, tables, but never on the mat, until one fine afternoon, according to Henry, whose parents, on hearing the unlikely news, asked in disbelief....
I don't know whether punctuation is part of grammar at all. Is it? I don't know whether, say, "Someone had a cat that would happily sit on carpets chairs tables but never on the mat" is ungrammatical. I know it's wrong; but whether that 'wrongness' lies in it being ungrammatical, I don't know. But I'm not sure I'd take North Somerset performance poet Melanie Branton's opinion as definitive.