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Using a spherical Bohr-radius volume for the atom, the calculations below give a cutoff electromagnetic <vacuum energy density> of about $6.2\times10^{15}\,\mathrm{J\,m^{-3}}$, compared with a <hydrogen> binding-energy density of about $3.5\times10^{12}\,\mathrm{J\,m^{-3}}$. \b[The cutoff vacuum estimate is about $1.8\times10^3$ times larger.] Alternative reasonable definitions of atomic volume change this ratio by a factor of a few, not the main order-of-magnitude conclusion.
The cutoff result is a bare zero-point mode sum. It is not a classical radiation bath incident on the atom, nor a calculation of an observable absolute vacuum energy without a subtraction or <renormalization> prescription.
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