Mobile button style navigation

I'm updating the format of our conference mini-sites to function like our mobile apps on phones and still function like normal websites otherwise. Along with the top nav bar I will have a mobile icon nav bar for the secctions.

What is the best way to build that?

Should I set it up using a list <ul><li> then setting a flex direction of row or would it be better to set it up in columns with <section><div>, or a completely different way?
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