Think it is going to be difficult to get a local website/host that offers that. As it takes a lot of time and effort to setup an online store.
But why don't you use a script like oscommerce or opencart. You can even make an online store with wordpress these days. Its really not that difficult to create an online store with these scripts. And you will be in full control with everything.
Here is a list of script scripts that you can use for your online store:
https://www.softaculous.com/softaculous/apps/ecommerce
It depends what you want to do on the online store. Eg. how many items do you have, how do you want to collect payment, are you selling items or services, etc. There is also lots of videos on youtube that will teach you the basics of the script.
Let me know if you want some help or want to check out a specific script. I can install it on my test website and let you have to login details to play around on it and have look.
Oh, and if you use a nice template for the script which is responsive, then its not even necessary to have an app, because the user will be able to open your website on his/her mobile/tablet. (Responsive-> Responsive Web design (RWD) is a Web design approach aimed at crafting sites to provide an optimal viewing experience—easy reading and navigation with a minimum of resizing, panning, and scrolling—across a wide range of devices (from mobile phones to desktop computer monitors)
