We investigate the effect of disorder on the heat transport properties of the $S=tfrac{1}{2}$ Heisenberg chain compound Sr$_2$CuO$_3$ upon chemically substituting Sr by increasing concentrations of Ca. As Ca occupies sites outside but near the Cu-O-Cu spin chains, bond disorder, i.e. a spatial variation of the exchange interaction $J$, is expected to be realized in these chains. We observe that the magnetic heat conductivity ($kappa_{mathrm{mag}}$) due to spinons propagating in the chains is gradually but strongly suppressed with increasing amount of Ca, where the doping dependence can be understood in terms of increased scattering of spinons due to Ca-induced disorder. This is also reflected in the spinon mean free path which can be separated in a doping independent but temperature dependent scattering length due to spinon-phonon scattering, and a temperature independent but doping dependent spinon-defect scattering length. The latter spans from very large ($>$ 1300 lattice spacings) to very short ($sim$ 12 lattice spacings) and scales with the average distance between two neighboring Ca atoms. Thus, the Ca-induced disorder acts as an effective defect within the spin chain, and the doping scheme allows to cover the whole doping regime between the clean and the dirty limits. Interestingly, at maximum impurity level we observe, in Ca-doped Sr$_2$CuO$_3$, an almost linear increase of $kappa_{mathrm{mag}}$ at temperatures above 100 K which reflects the intrinsic low temperature behavior of heat transport in a Heisenberg spin chain. These findings are quite different from that observed for the Ca-doped double spin chain compound, SrCuO$_2$, where the effect of Ca seems to saturate already at intermediate doping levels.
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