Yes, absolutely you can create application for Facebook without using any SDK..
SDK is just a software development kit (SDK or "devkit") it's typically a set of software development tools that allows for the creation of applications for a certain software package,
Facebook SDK made your work much easier because they give you ready made functions to call Facebook API like PHP sdk ,C# sdk.. etc
If you want to create application without using it .. then you have to request Facebook Graph API directly ..
For e.g you need to get page information using JS SDK ..you need to pass only name/id of the page in JS SDK
FB.api('/cocacola', function(response) {
console.log(response);
});
but if you want to go without SDK, then you need to call http://graph.facebook.com/cocacola
now you need to parse JSON
then retrieve data based on the language you're working on