  Background: Shape coexistence in heavy nuclei poses a strong challenge to
state-of-the-art nuclear models, where several competing shape minima are found
close to the ground state. A classic region for investigating this phenomenon
is in the region around $Z=82$ and the neutron mid-shell at $N=104$.
  Purpose: Evidence for shape coexistence has been inferred from $\alpha$-decay
measurements, laser spectroscopy and in-beam measurements. While the latter
allow the pattern of excited states and rotational band structures to be mapped
out, a detailed understanding of shape coexistence can only come from
measurements of electromagnetic matrix elements.
  Method: Secondary, radioactive ion beams of $^{202}$Rn and $^{204}$Rn were
studied by means of low-energy Coulomb excitation at the REX-ISOLDE facility in
CERN.
  Results: The electric-quadrupole ($E2$) matrix element connecting the ground
state and first-excited $2^{+}_{1}$ state was extracted for both $^{202}$Rn and
$^{204}$Rn, corresponding to ${B(E2;2^{+}_{1} \to 2^{+}_{1})=29^{+8}_{-8}}$
W.u. and $43^{+17}_{-12}$ W.u., respectively. Additionally, $E2$ matrix
elements connecting the $2^{+}_{1}$ state with the $4^{+}_{1}$ and $2^{+}_{2}$
states were determined in $^{202}$Rn. No excited $0^{+}$ states were observed
in the current data set, possibly due to a limited population of second-order
processes at the currently-available beam energies.
  Conclusions: The results are discussed in terms of collectivity and the
deformation of both nuclei studied is deduced to be weak, as expected from the
low-lying level-energy schemes. Comparisons are also made to state-of-the-art
beyond-mean-field model calculations and the magnitude of the transitional
quadrupole moments are well reproduced.
