That's a great question!

In the last few years there is plenty of research on the subject - and a few (interesting) open questions.

Before Kawamta's result - note that toric manifolds satisfy $rk(K(X))= \chi(X)$ (this is a result on the fan), so conceptually they are a good candidates to have a full exceptional collection (as Kawamata's result shows), see Remark 2.3 in Uhera's work on derived categories of toric threefolds.

Kawamata's result (mentioned in the answer above) is very deep but the exceptional collections he constructs can be quite hard to track (the construction relies on various ideas from toric Mori theory). Pay attention that Kawamata's construction of e.c is not necessarily strong - which is a condition one typically wants an exceptional collection to satisfy.

If you will look at the first examples of (full strongly) exceptional collections discovered by Beilinson for $X=\mathbb{P}^n$ (which is, of course, toric) you will see that they are given by rather natural elements $\mathcal{E}_1 = \left \{ \mathcal{O},\mathcal{O}(1),...,\mathcal{O}(n) \right \} $ and $\mathcal{E}_2 = \left \{ \mathcal{O},\Omega^1(1),... , \Omega^n(n) \right \}$. In fact $\mathcal{E}_1$ is given by line bundles, that is elements of $Pic(X)$.

This led A. King to ask - which toric manifolds $X$ admit exceptional collections of line bundles in $Pic(X)$. For a while it was strongly believed that any toric manifold admits such a collection - but this was proved to be wrong by L. Hille and M. Perling.

In particular, this question (which toric admit e.c of line bundles) turned out to be quite an elusive question regarding toric manifolds. For instance - there is no clear algorithm that determines on the basis of the toric fan $\Sigma$ whether the corresponding manifold has such a collection.

On the other hand - there's plenty of research and results which shows that exceptional collections of line bundles for toric manifolds $X$ are a rather interesting class of exceptional collections - for instance just to mention a few - A. Bondal, Costa and Miro-Roig, H. Uehara, Borisov and Hua, Dey, Lason and Michalek, Bernardi and Tirabassi...

Personally, I am interested in connections between such collections - and ideas from mirror symmetry - they turn to be related to solutions of toric Landau-Ginzburg system (which are given in terms of the fan) - if your'e interested - there's some details here.