Does app store or Play Store allow apps that have a WebView only to my site?

Yes, you can published an app which contains just a WebView of your own website. Please note that according to Google Play Developer Program Policies, it's not allowed to publish apps that provide a WebView of a website you don't own or administer:

Do not post an app where the primary functionality is to:

Provide a webview of a website not owned or administered by you (unless you have permission from the website owner/administrator to do so)

Reference: http://play.google.com/about/developer-content-policy.html

According to Apple's App Store Review Guidelines, apps which are just web views are not allowed:

4.2 Minimum Functionality

Your app should include features, content, and UI that elevate it beyond a repackaged website. If your app is not particularly useful, unique, or “app-like,” it doesn’t belong on the App Store.

Reference: https://developer.apple.com/app-store/review/guidelines/#functionality

Yes You can but when you implement webview in your application few things must keep in mind Webviews and Affiliate Spam We don’t allow apps whose primary purpose is to drive affiliate traffic to a website or provide a webview of a website without permission from the website owner or administrator.

Here are some examples of common violations:

An app whose primary purpose is to drive referral traffic to a website to receive credit for user sign-ups or purchases on that website. Apps whose primary purpose is to provide a webview of a website without permission: Google play Developer Policy Center link