Your n for WGCNA is 336, not 6. Network construction pools all samples to estimate gene-gene correlations, and 336 is plenty (the authors' floor is around 15). So on the "can I", yes.

The repeated measures problem is real but it bites somewhere other than where you're worried. Network construction is fairly tolerant of it, since you're estimating a correlation structure rather than testing hypotheses. Where it matters is the module-trait step: feed 336 samples into moduleTraitCor as if they were independent and your p-values come out badly anticonservative, because your real independent units are individuals, not samples. Test the module eigengenes with a mixed model instead, something like ME ~ timepoint * group + (1|individual), and take your inference from there rather than from the correlation heatmap's p-values.

The other thing worth checking: in longitudinal data the between-individual baseline differences are often bigger than the treatment effect, so modules can end up mostly encoding individual identity rather than vaccine response. Plot a module eigengene coloured by individual and see whether it stratifies that way. If it does, consider centering each gene within individual before building the network, so it reflects change rather than absolute level. Whether you want that depends on your question -- centering finds co-varying response programs, not centering finds programs that differ between animals.