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What might be the physical intuition behind the fact that we don't need to renormalize the field/wave function at one-loop level in $\phi^4$ theory? I justify the field renormalization thinking "the bare field can't connect the vacuum to multi-particle states which are indeed present there due to interaction." Going with this "intuition" I don't see why would we not renormalize the field at one-loop as whenever there is loop there is this possibility of connecting a state with multiparticle states?

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There is no intuition, really. In this particular theory, the first order contribution to the wave strength renormalization just happens to vanish.

Why does it vanish? Well, this constant comes from the kinetic term, so it is associated to the momentum of the diagram. And, to first order, the only diagram is the slug

enter image description here

It is clear that the diagram is momentum independent, by conservation of momentum at each vertext. So there is no renormalization to this order. The next order, the sunset

enter image description here

does involve momentum so here the field does get renormalized.

There is no a priori reason why this should have been this way.

If you insist in finding something more, the best I can offer is the following. The slug is a tadpole, meaning that this diagram includes a self-contraction. As such, the diagram can be made to vanish by a proper ordering prescription: normal ordering. But normal ordering involves a restructuring of the potential energy, not the kinetic energy (cf. this PSE post). In this sense, you could have predicted that there is no wave renormalization to first order, because in the normal-ordered theory there are no first order diagrams at all. But this is no intuition either...

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