Is it required to use choice principles to prove that any linearly independent set can be extended to a basis?

Just the existence of a basis for every vector space is enough to conclude the axiom of choice, as one can extend the empty set (or any singleton, except $\{0\}$).

For the finite-dimensional case: If the linearly independent set is not a basis, there is a vector that it does not span i.e. that is linearly independent of it. Thus, we can increase the size of the linearly independent set by adding this vector to it. By induction, we can increase its size as long as it is not a basis. Since the vector space is finite-dimensional, this process must terminate in a basis.