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Here is my constructor code:

vertex_array(float* &vertex_buffer, std::string& texture_file);

Here is my main application code:

float cube1 = 0.5f, -0.5f, 0.5f, 1.0f, 0.0f,
 0.5f, -0.5f, -0.5f, 1.0f, 1.0f,
 -0.5f, -0.5f, -0.5f, 0.0f, 1.0f,
 -0.5f, -0.5f, 0.5f, 0.0f, 0.0f,

 0.5f, 0.5f, 0.5f, 1.0f, 0.0f,
 0.5f, 0.5f, -0.5f, 1.0f, 1.0f,
 -0.5f, 0.5f, -0.5f, 0.0f, 1.0f,
 -0.5f, 0.5f, 0.5f, 0.0f, 0.0f ;

vertex_array va_1(cube1, "blocks.png");

I am getting the error: no instance of constructor "vertex_array::vertex_array" matches the argument list argument types are: (float[40], const char[11])

I have spent around an hour looking for the problem yet I cannot find anything wrong with the code. I even looked at another example that passed in the exact same argument type but to a parameter of type const void* and it worked on the example however not for me. What is my problem?

The first argument you give to the constructor is a pointer to a number of floats and the second is a const char* which will be implicitly converted to a const std::string, so you need to change the signature accordingly:

vertex_array(float* vertex_buffer, const std::string& texture_file);

You could also remove the reference declaration for texture_file to make it valid but if you are later going to instantiate a vertex_array using a real std::string, it would result in an unnecessary copy construction, so go for the const reference.

If you have different sizes of vertex_buffers, you need to supply the size as an argument too or consider using std::initializer_list or a container to keep track of the number of elements.

Example:

vertex_array(std::initializer_list<float> vertex_buffer, const std::string& texture_file);

vertex_array va_2(
 0.5f, -0.5f, 0.5f, 1.0f, 0.0f,
 0.5f, -0.5f, -0.5f, 1.0f, 1.0f,
 -0.5f, -0.5f, -0.5f, 0.0f, 1.0f,
 -0.5f, -0.5f, 0.5f, 0.0f, 0.0f,

 0.5f, 0.5f, 0.5f, 1.0f, 0.0f,
 0.5f, 0.5f, -0.5f, 1.0f, 1.0f,
 -0.5f, 0.5f, -0.5f, 0.0f, 1.0f,
 -0.5f, 0.5f, 0.5f, 0.0f, 0.0f, "blocks.png");

I think you may be using your address of operators & incorrectly.

Remove Them both from the declaration and try again.

Using them for the array is pointless and literally cancels out what you are trying to achieve. The second argument is also incorrect because you are trying to get the address of an r-value.