std::vector of objects and const-correctness

I'm not sure why nobody said it, but the correct answer is to drop the const, or store A*'s in the vector (using the appropriate smart pointer).

You can give your class terrible semantics by having "copy" invoke UB or doing nothing (and therefore not being a copy), but why all this trouble dancing around UB and bad code? What do you get by making that const? (Hint: Nothing.) Your problem is conceptual: If a class has a const member, the class is const. Objects that are const, fundamentally, cannot be assigned.

Just make it a non-const private, and expose its value immutably. To users, this is equivalent, const-wise. It allows the implicitly generated functions to work just fine.

An STL container element must be copy-constructible and assignable¹(which your class A isn't). You need to overload operator =.

_{1 : §23.1 says The type of objects stored in these components must meet the requirements of CopyConstructible types (20.1.3), and the additional requirements of Assignabletypes}

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EDIT :

Disclaimer: I am not sure whether the following piece of code is 100% safe. If it invokes UB or something please let me know.

A& operator=(const A& assign)
{
    *const_cast<int*> (&c)= assign.c;
    return *this;
}

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EDIT 2

I think the above code snippet invokes Undefined Behaviour because trying to cast away the const-ness of a const qualified variable invokes UB.

You're missing an assignment operator (or copy assignment operator), one of the big three.