Of course if you insist on condition $\langle \mathbf{x},\mathbf{x}\rangle > 0$, and not merely $\langle \mathbf{x},\mathbf{x}\rangle \ne 0$, then you must have an order.  

Let $F$ be a formally real field.  Then
$$
\langle \mathbf{x}, \mathbf{y}\rangle = \sum_{j=1}^n x_j y_j
$$
can be a reasonable inner product on $F^n$.  According to an ordering for $F$ (indeed, any ordering, since there may be more than one) we have $\langle \mathbf{x}, \mathbf{x}\rangle > 0$ if $\mathbf x \ne \mathbf 0$.  

Another part that you quote is what would be required for metric completeness.    Do you want that? If $F$ is a proper subfield of $\mathbb R$, then even the one-dimensional space is not complete.  

Something weaker than completeness will be enough to carry out the Gram-Schmidt process.  It requires only that square-roots of $\langle \mathbf{x}, \mathbf{x}\rangle$ exist.