Um, no. At least, not this writer. I have tried carrying around a notebook, and it's jolly useful for jotting down a shopping list. I've also added brilliant ideas for books and articles, and amazing titles. I've dutifully written down hilarious snatches of overheard conversation. And then either I lose the notebook or I forget about it.
If I do find it, I discover that the snatches of conversation make no sense out of context, or simply aren't funny. The brilliant titles aren't that brilliant, and nor are the articles and book ideas.
For me, carting a notebook around is adding yet another bit of stuff to my handbag. That's not to say you shouldn't do it, because if it works for you then yippee and hooray. But it doesn't work for me.
All writing advice should be taken with a hefty dollop of salt. Try everything, but don't feel you have to stick with it if it doesn't work for you. Some of us need detailed plots before starting, others write into the blue. Some of us need silence, others want noise when they write. Some of us write thousands of words a day, others struggle to complete a few hundred.
Who cares how you got there and what tools you used? There isn't a 'right' way to write, there is only your way. All that matters is you get there in the end, notebook or not.