I think it may be possible, but you will have to consider that your diet will probably affect the towns you'll be able to stop in, as well as how much weight you'll be carrying with you at any given moment. The Primitivo is my probably my favorite Camino, but is also one of the routes with fewer provisions. That said, with some careful planning, I'm sure you can make it work. You will need to plan on making the majority of your meals to stick to a specific diet. I'd also consider carrying with you a lightweight bowl/spoon/fork and picking up a small pocket knife upon arrival (I do this even without a special diet!)--which will make it much easier for you to be flexible. You'll also want to keep in mind that stores are typically closed on Sundays, so you'll need to plan a day ahead. For planning purposes, you may want to pick up a book or at least go carefully through one of the camino sites (like
http://caminodesantiago.consumer.es/) and plan your itinerary so that you'll have the facilities you need along the way. Good luck!
If you begin in Oviedo, you will easily be able to find shops there, and the municipal albergue has a kitchen.
If you break down your stages:
1. Oviedo-San Juan de Villapañada(29.5km) (which is 4km outside of Grado), you'll want to restock in Grado, the only sizable town you'll pass through during the day. The Albergue in San Juan has a kitchen, but no provisions, so you'll need to carry supplies from Grado (but, again, this is not far!). Keep in mind that the grocery stores are closed on Sundays (though I believe it's a market day) and on Mondays in Grado.
2. San Juan - Salas (17.5km)-- You'll be able to resupply in Cornellana (8km), and again in Salas (9.5km from Cornellana).
3. Salas - Tineo (19km)--There's not a lot along the way, but there is a small grocery in La Espina, halfway between the two, and Tineo has multiple grocery stores.
4. Tineo-Pola de Allende (25km)--You'll again have provisions both in Tineo and Pola. There is a small grocery in Campiello, and Casa Herminia also sells provisions. *NB. I would not recommend doing the Hospitales Variant, which is beautiful, but much sparser provision-wise, as it skips Pola de Allende, the only main town on this section of the route.
5. Pola-Grandas de Salime (32.5km) This is a very long day, but allows you to stay in Pola and Grandas, the two sizable towns along this part of the route. You'll be able to get some provisions in Berducedo (12km into the walk, grocery located in the bar), but it's fairly limited. Grandas de Salime will have more options for you.
6. Grandas - Padrón (26.5km) This walk is fairly stark provisions-wise, so you'll want to plan for the majority of the walk in Grandas. The albergue in Padròn is located just 1.5 km (1km via highway) past Fonsagrada, a large town with multiple markets. The albergue in Padròn has a kitchen. (it's an easy walk back into Fonsagrada from Padròn.)
7. Padròn - Castroverde (31km) There will be grocery stores in Càdavo Baleira (22km into the walk) and again in Castroverde, the albergue in Castroverde has a kitchen.
8. Castroverde - Lugo (21km) There's little on the walk, but you'll be set once you arrive in Lugo.
The stretch from Lugo-Melide may be the most difficult, as, if you follow the route through San Roman and As Seixas, there are no grocery stores from Lugo-Melide (about 50km). However, if you walk from Lugo to Palas de Rei (35km), you'll be able to stock up adequately in Palas de Rei (which has multiple grocery stores, and is where the route joins the Camino Francès). You'll see the turn-off marked for Palas in yellow spray paint--the route just follows a small country road, the CP-29-01. From Palas, you should be set to Santiago, as the route passes through a number of larger towns (Melide, Arzùa, and Pedrouzo all have groceries), and soon after that you'll arrive in Santiago!