Ciao Dwrose4 and welcome to the forum!
I think you can most definitely find what you're looking for - there are many agriturismos that meet the parameters you're looking for. I'm surprised you can't find much on Booking.com, so many are on that platform (as we are an affiliate partner, I'll link a few below).
First, the areas: since Val d'Orcia is pretty close to Siena and to the south of the city, I'd suggest your other base be to the NORTH of Siena in the Chianti area. That way you can use that base for 3 days and visit: all the hilltop towns such as Greve, Radda, Castellina, Panzano and Gaiole, as well as San Donato in Poggio and make your way to San Gimignano and even Volterra - with a visit to Siena either while you're here or on your way down to your second base (or the other way around, doesn't matter which one you do first).
In the Val d'Orcia area, you can visit Montepulciano, Pienza, San Quirico d'Orcia, Montalcino, Arezzo and Cortona.
Just make sure to include Siena in the stay where you are longer - it can be easily reached from both Chianti as well as Val d'Orcia.
Now as far as agriturismos go - look for those that are rented by the day/by a room and they will automatically offer breakfast. They are essentially countryside hotels/B&Bs, depends on their size. You want to stay away from any that only offer weekly rentals.
Now that I've offered tips for location and type, I can suggest some specific ones we have reviewed ourselves on tuscanyaccommodation.com -these are the ones in the Val d'Orcia area:
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Myrtus apartment at Podere Cunina is perfect for couples and located very centrally - only "defect" from your point of view is that is is an apartment and thus no breakfast is included, but take a look at the pictures of the location!
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BelSentiero - this is south of San Quirico d'Orcia so very central and also closer to the hot springs areas of Bagno Vignoni and San Filippo, two places I highly recommend if you want to try the hot springs (both outdoors free or indoors paid).
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Agriturismo Castello di Spedaletto has great reviews for its breakfast and the quiet and peace
You can check more options in Val d'Orcia near Pienza by
plugging your dates here.
As for the Chianti area near Siena, we highly recommend
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Romitorio di Serelle - this might actually entice you to stay put for the entire 7 days in one place, it is pretty central and just surrounded by gorgeous vineyards.
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Tenuta Quadrifoglio has breakfast included for the rooms.
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Palazzo Malaspina has breakfast included and is located in the middle of the charning San Donato in Poggio hamlet, very close to all the Chianti towns and San Gimignano.
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Villa Fillinelle also has breakfast as well as options with kitchen, so you could do a mix of both.
You can check out
more options for Chianti here. Let me know if there are any in particular you like more than others, so I can think up of other options to recommend. I really like
Dianella too, but it west of Florence so further out of the area, but as a winery, it offers a great countryside experience, with restaurant included.