Numerical precision in sum-of-squares method?
I think my answer is probably insufficient, but it remains for completeness' sake (although see Boaz's comments below for probably a better answer)

When we limit ourselves to boolean variables, the claim can be seen when $(x_i^2-1) \in E$ for all $i \in[n]$ with the observation that degree $2n$ pseudo-distributions are actual distributions, that is, suppose you have a pseudo-distribution $\mu(x)$ over solutions $x$ of your polynomial equalities $E$ satisfying:

$\sum_{ x \in \{-1,1\}^n} \mu(x)$ and $\sum_{x\in\{-1,1\}^n} \mu(x) p^2(x)\ge0$ for all polynomials $p$ with degree at most $n$

But degree $n$ polynomials include the indicator polynomial (for example, $x_1 = 1, x_2=-1, x_3=1$ has $2^{-3}(1+x_1)(1-x_2)(1+x_3)$ which is all-zero elsewhere and 1 on that assignment). So $\mu(x) \ge 0$ for all $x\in\{-1,1\}^n$, so we conclude $\mu$ is an actual distribution over the solutions of $E$. Degree $\ell$ pseudo-distributions can be found by using semidefinite programming to find an associated degree $\ell$ pseudo-expectation operator in $n^{O(\ell)}$ time, so we can find the actual distribution $\mu$ in time $n^{O(n)}$ by using that pseudo-expectation (now an actual expectation) to find all the moments of $\mu$.

So, if $|E| = O(1)$, then you can find a distribution of solutions to $E$ in $O(1)$ time. Of course, brute force search guarantees the same. 

However, if the solutions are not necessarily boolean, then degree-$2n$ pseudo-expectations are not sufficient to find a distribution over solutions. As can be seen above, the proof that degree $2n$ pseudo-distributions are actual distributions depends on the fact that degree $n$ polynomials are sufficient to 'pick out' individual assignments, which is not true more generally. Another way of viewing it is that boolean-variable polynomials are considered $\mod(x_i^2)$, so the degree of every monomial is at most $n$.

For example, one could consider replacing every binary variable with a 4-ary variable, say by including $(x_i^2-1)(x_i^2-4) \in E$. Then you would have to have a degree $4n$ pseudo-expectation in order to guarantee recovery of a distribution over solutions.

Now, for theoretical guarantees, it seems like approximating a root of a system of polynomals is also known as Smale's 17th problem, and apparently there is a randomized (Las Vegas) polynomial time algorithm that solves this - see http://arxiv.org/pdf/1211.1528v1.pdf. Note that this seems to be in the Blum-Shub-Smale model, so real operations are the primitive. I am not sure if this gives the guarantee that you need.