I don't know the place specifically but ime you do need a car in Sardinia, not least because there are practically no trains and public transport is a bit here and there. I went once without a car and we were stuck in our isolated holiday village and going out of our minds with boredom. Even getting to the nearest town was a nightmare.
However ,it does depend on whether you're quite happy staying in one place or would prefer to explore. I presume Liscia di Vacca is a town in itself and not a holiday village? If so, you'll probably be OK with finding places to eat out locally, but if it's tiny you may get bored. How long will you be going for?
I've just seen, it's near Porto Cervo. Very nice but quite pricey, depending on the time of year. If you had a car you could go and visit places like La Maddalena, but there might be a boat trip direct from the port. There's also l'isola dei Gabbiani a bit further north, plus other less frequented beaches that are worth a visit. Maybe keep the option of getting a car for the odd day?
IIRC there is a bus service that does Porto Cervo/Arzachena to Palau etc., but it is quite infrequent.
There'll be lots of trattorie and pizzerie, the latter being the cheap restaurant option in Italy. Go to the less exclusive towns like Palau or Arzachena and you should find something quite easily. Avoid anything overtly touristy or too snazzy/tarted up (eg. Flavio Briatore's Millionaire club) and avoid Porto Cervo itself like the plague. For food shopping if you self cater there's a large shopping centre with a big supermarket on the road to Olbia if you want to avoid overpriced resort shops. Olbia has also got a few places to eat out and more choice, being a larger town.
Eating out in Italy is loads cheaper than in the UK in general though, so you may find it quite cheap anyway. I'm no longer used to London prices so see things at a different level.
Another cheap option if the weather is nice in the evening is to go to one of the many take away places they have in the squares. . You can sit down on a wall + eat and watch the world go by You can get yummy pizza by the slice, various anti pasti type stuff + even horse steak in a bun if you are feeling adventurous