For a symmetric matrix $A$, I am aware that eigenvectors $v_1, \dots, v_n$ with the same eigenvalue $\lambda$ are linearly independent but not orthogonal.
The spectral theorem states that any $p \times p$ symmetric matrix has $p$ orthonormal eigenvectors. I do not understand how both these statements are correct when an eigenvalue can correspond to multiple eigenvectors and thus these eigenvectors can only be linearly independent with one another, and not orthogonal?
You can get an orthonormal basis from the $v_1,\dots,v_n$ using Gram-Schmidt, say. Then, if $v_1',\dots, v_n'$ is such a basis, it is easy to see that each $v_i'$ is still an eigenvector for $\lambda $. This basis satisfies the requirements of the spectral theorem.