My first thought.
Dépanneur was my second thought, although I’m on the other wrong coast.
We have a Circle K on the highway near us. I only like to stop there so that I can whisper “strange things are afoot at the Circle K”
In English and unless I know its actual name, “convenience store”.
Yup, in Michigan they are Party Stores, and (at least around here) always sell beer, wine, and liquor as well as bread & milk & chips & candy.
Unless they also sell gas, in which case they are gas stations, and typically only sell beer & wine, no liquor.
I had no clue that “party store” wasn’t a general term until I moved away from Michigan, and whenever I said I was heading to the party store, they expected me to come back with balloons and decorations, not a six-pack and a loaf of bread.
We used to use zip and rip, but now just call it the Tower or Bobs, depending on which one we need.
I generally refer to such places as a corner store. Unless, for some reason, I feel the need to mention that I intend to get gas as well, then it becomes a gas station to differentiate between such places as CVS or Walgreens.
That’s what we called the one on the corner near my mom’s house, in Seattle. It changed hands so often, we didn’t bother to keep up with the actual name.
Around here, I call them by name – Ampride, Casey’s, or Kum 'n Go.
For real? 
They’re all over the Midwest.
They even sponsor things.
(Never let teens pick the sponsor.)
I think in Japan this type of place is called a “conveni” …
Yup, party in a store, leave happy.
Where I come from, they were all veepees. Guesses?
The ones in Germany on the Army posts were called “Foodland.”
~VOW
[hijack] I was in Target yesterday and they had these weird Target Bodega signs up, with arrows pointing into the bowels of the store. I didn’t follow to see WTF they were trying to achieve with a bastardized corporate “bodega” in a hugely Hispanic neighborhood. I hope, for their sake, it was pointing to the wine department, not a section of dusty toiletries, Bimbo bread and Saint candles.[/hijack]
I can’t pass a Kum 'n Go without giggling.
Anyhow, our local go-to we call “the 7-11”, because it’s a 7-11.
If I’m at someone else’s house and don’t know the name of what’s nearby, I’d ask if there was a “drugstore or convenience store” nearby, and hope to hear Walgreens or CVS. Still overpriced for bread and milk, but not quite so much as 7-11 or one of the gajillion actual bodegas around here.
Growing up, everyone called the main local store the Dairy. I’m not really sure why. Maybe the building was a dairy once. That name stuck for all the convenience stores for me.
I miss that word. 
If there was a store like that nearby that I went to often, I think I would fall into calling it the dep, but the grocery store is so close that I don’t really need to go elsewhere.
Corner store or convenience store. The closest one to my house is on the corner, so I guess I would refer to it that way, except that it’s a Hasty Mart, and I always change the H to an N and call it that.
By name, or gas station if the name isn’t known. And BTW: my local, Kwik Trip, has the best milk and bread prices in town. $1.09 for half gallon of 1%B milk, and 2/$1.39 loaves of bread.
Mostly, I call it the gas station, because that’s where I pick up cokes and smokes close to home.
Other times I’ll refer to it by name (Flash Market) or use another local chain (Kwik-Sak) as a sort of generic.
The store’s actual name, which around here is inevitably “QD”, for Quality Dairy. If it sells mostly liquor, it becomes a “party store”.