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Consider a calculation of photon vacuum polarization in scalar QED. The 1PI diagram which people calculate is this one,
But in scalar QED the seagull vertex is also possible
Then why not include a diagram like this one,
Which would then make the propagator sum to go something like this,
and so on.
Now my question is am I correct to add such a diagram? If not why? Also if indeed such terms are added in a full calculation please refer to articles which take these contributions into account.
That's a completely valid diagram, but it's higher order in perturbation theory. The 3-point vertex comes with a factor of $e$, and the 4-point vertex comes with a factor of $e^2$, so your diagram is order $e^4$, which is higher than the order $e^2$ contributions calculated in standard textbooks. Note that for the purposes of computing corrections to the propagator in this theory, "order $e^2$" is synonymous with "one loop", and "order $e^4$" is synonymous with "two loop".
Two loop corrections can be found in various papers. For example, this paper claims to calculate two-loop corrections in scalar QED using dimensional regularization.
