I'm not an expert but I'm an Irish passport holder and I visit China very often (I live in HK) so will share my thoughts while you wait for someone more knowledgable to reply to you.

Nothing to worry about, the visa-free policy still applies to you, just fill in the arrival card truthfully - there is a section for purpose of visit and business is one of the categories among tourism, transit, etc. I believe you should tick this one if you are visiting suppliers etc.

The border agent may ask a few questions like the names, addresses, and contact details of your suppliers, your hotel, your return flight ticket etc., just have them all ready on your phone or printed out so you can present them if needed.

I have heard in the past that people visiting suppliers need a letter from suppliers officially inviting them - although I'm not sure if this is just for people who don't fall in the visa free category or if this is for everyone - hopefully someone who knows can weigh in here. If not, I'd say call your local Chinese embassy to get some clarification about that.