Keep in mind most towns will have a florist and a hardware store but they will be small, family-owned enterprises.... so it really depends if this will be enough for your needs or will need something like OBI, which is a large store that contains most of what you can imagine regarding hardware as well as home items.
I found La Magona on Google Maps -
https://goo.gl/maps/5P8PoVXEkNs - classified as a hardware store (ferramenta) but not sure whether it is still open. The website that is linked on the profile is definitely not them.
Also found a florist -
https://goo.gl/maps/ueU8otiAmm12
But keep in mind shops close and change, Google Maps could be outdated.... and then August is August. While Chianti is in high season and full of visitors, many shops remain open but most small shops will still close around the 15th of August to honor Virgin Mary so make sure to plan accordingly. I would suggest you scout around town on your first days before or after dinner, with a gelato in hand and enjoy a
passeggiata and check out whether you find these shops, or just go ahead and ask at a local bar since they will know where to send you! ;-)
Finally, as Donna has suggested, most small towns have a day of the week with a market that takes over and there are many stands and together they offer a bit of everything - make sure to visit the one in Castellina to see what they have.