Here is a counterexample: Take $n=2$ variables $X$ and $Y$. Let $L_1,\dots,L_4$ be the polynomials $1\pm X$ and $1\pm Y$ and $P=2-X^2-Y^2$. Assume we could write $P$ as the sum of a globally nonnegative quadratic polynomial $S$ and nonnegative linear combinations of the $L_i$ and $L_iL_j$. Now $P$ vanishes at the four points $(\pm 1,\pm 1)$ and each $L_i$ is nonnegative at $(\pm 1,\pm 1)$. Therefore $S$ vanishes also at $(\pm 1,\pm 1)$. But being a nonnegative quadratic polynomial, $S$ is a sum of squares of linear polynomials which all have also to vanish at $(\pm 1,\pm 1)$ and therefore are identically zero. This shows that $S$ is the zero polynomial. Now notice that each of the $L_i$ and $L_iL_j$ is strictly positive on at least one of the points $(\pm 1,\pm 1)$ where $P$ vanishes. Since $P$ is a nonnegative linear combination of the $L_i$ and $L_iL_j$, this shows that $P=0$.
If the set defined by the simultaneous inequalities $L_i\ge0$ has non-empty interior, then the convex cone of quadratic polynomials which can be written as a globally nonnegative quadratic polynomial $S$ and nonnegative linear combinations of the $L_i$ and $L_iL_j$ is closed. In fact, this follows from a much more general result on truncated quadratic modules, see e.g. the book of Marshall cited below (Lemma 4.1.4). This implies that, in the above counterexample, even $P+\varepsilon$ for small $\varepsilon>0$ will fail though this polynomial is strictly positive on the set defined by the inequalities $L_i\ge0$.
However, there are a lot theorems going into the direction of what you want. You might want to have a look at the following books...
- Marshall: Positive polynomials and sums of squares
- Prestel: Positive polynomials
- Bochnak, Coste, Roy: Real algebraic geometry
- Basu, Pollack, Roy: Algorithms in real algebraic geometry
- Knebusch, Scheiderer: Einführung in die reelle Algebra
- Andradas, Bröcker, Ruiz: Constructible sets in real geometry
...and the following articles...
- http://homepages.cwi.nl/~monique/files/moment-ima-update-new.pdf
- http://www.math.uni-konstanz.de/~schweigh/publications/purestates.pdf
- http://www.math.uni-konstanz.de/~schweigh/publications/sosdualsdp.pdf
Also the so-called "S-procedure" could be of interest for you.