Can you complete a basis in polynomial time?
The result of the Gram-Schmidt can be expressed in determinantal form, see Wikipedia. This shows that the output of the Gram-Schmidt process is polynomial size. This suggests that if you run the classical Gram-Schmidt process, then all intermediate entries are also polynomial size (even in LLL, all intermediate entries are polynomial size). However, even if it is not the case, then using efficient algorithms for computing the determinant (see my other answer), you can compute the Gram-Schmidt orthogonalization in polynomial time.