You could check SharePoint REST API to make ajax request and perform your expected behavior.

https://msdn.microsoft.com/en-us/library/office/dn499819.aspx#bk_WebGetSubwebsFilteredForCurrentUser

You could also take time to learn how to use JSOM API to do the same think (but too heavy to implement if you have not a friendly framework :)

https://msdn.microsoft.com/en-us/library/office/jj246242.aspx

Hope it helps :)