In many literature about elliptic regularity on convex domains, they impose that the domain should be polygonal. (In such cases, the inequality below holds)
However, it seems to me that the polygonal assumption is given to certify regularity of variational(weak) solution rather than inequality.
Question) For convex domain $\Omega$ with piecewise smooth boundary, let $u$ be an $H^2$ solution of $\Delta u=f$ with $u|_{\partial\Omega}=0$, does the following 'classical' inequality hold?
$$\exists C\,s.t.\,\|u\|_{H^2}\le C\|f\|_{H^0}$$
I am assuming by $H^{0}$ you mean the $L^{2}$ norm. If that is the case, then the answer to your question is yes for piecewise smooth Lipschitz boundary. In fact, any solution in $H_{0}^{1}(\Omega)$ solution will be $H^{2}$ with the above estimate in that case. Also, the hypothesis is being piecewise smooth is rather irrelevant here, the only important assumptions are being Lipschitz and convex. Convexity is absolutely essential and can not be dropped, as one can easily find counterexamples in a circular arc of angle greater than $\pi$ (thus nonconvex) even with $f=0.$ The Lipschitz condition can probably be weakened somewhat. You can start from P. Grisvard's classic book "Elliptic problems in nonsmooth domains" and go from there.