Peroxyacetic acid, $\ce{CH3C(O)OOH}$ has its anion form when a proton is detached, like $\ce{CH3C(O)OO-}$.
I think it can have two resonance forms like I drew (even though they have several charge separations, they're at least possible), but several websites address that it cannot have any resonance structures.
When you are considering resonance structures, you have to maintain certain aspects of the molecule; you have to maintain molecular formula/connectivity, number of electrons, and net charge. You maintain the molecular formula and number of electrons throughout your proposed structures, but what you fail to account for is net charge. In the true structure, there is a net charge of -1 resulting from the rightmost oxygen. In your structures, one example has a net charge of -2 while the other has -3. Here is a good source with the rules for resonance structures.
With that information, you can dismiss the resonance structures as invalid from the outset. This is because a resonance structure is the result of molecules "spreading" their electrons to achieve a more stable state; net charge will not get bigger. When you make the net charge bigger, as you have done, you make the molecule less stable, as there are now more lone pairs and charge is less evenly distributed. Furthermore, carbon will not pull an electron away from oxygen as you have done in your bottom drawing, at least in normal conditions; oxygen has a much higher electronegativity than carbon and will hold onto its electrons too tightly for carbon to remove them like that.
For these reasons, paracetic acid's conjugate does not show resonance because there is no resonance structure that follows the rules of resonance; the potential resonance structures you have considered are destabilizing, which is the opposite of the effect of resonance (resonance will always make a molecule's structure more stable).
Before we suggested any resonance structures for the conjugate base of peracetic acid (PAA), we have to prove it physically exists. The $\mathrm{p}K_\mathrm{a}$ of peracetic acid is found to be 8.3 (Ref.1). According to Ref.1, PAA is spontaneously decomposed to products given in equation $(1)$ in the $\mathrm{pH}$ range of 5 to 10: $$\ce{2CH3C(O)O-O-H -> 2CH3C(O)OH + O2} \tag1$$
The authors have shown that the maximum rate have occurred at $\mathrm{p}K_\mathrm{a}$ of PAA, and the mechanism of decomposition is gone according to following figure using labelled $\ce{CH3C(O)O^{18}-O^{18}-H}$.
Accordingly, nucleophilic peroxy $\ce{O-}$ would attack the electrophilic carbonyl carbon of acetic part of another molecule to give the product as shown. The mechanism was confirmed by finding 87% of decomposed oxygen has been in $\ce{O^{18}-O^{18}}$ formula. However, a different group (Ref.2) has proved that there is no peroxy $\ce{O-}$ exists during the decomposition during the $\mathrm{pH} \le 7.5$:
[...] in the preparation of PAA, the $\ce{H+}$ concentration in the system is usually more than $\pu{0.1 mol L−1}$. It indicates that PAA is present as molecular form and no peraetic anion is found in the reaction solution.
Thus, the reaction mechanism of PAA spontaneous decomposition in an acid system can be predicted as two-step reaction. Step 1 is as follows:
The protonated form $\mathrm{F_1}$ can be attacked by another unprotonated molecule in step 2 to give the active intermediate as follows:
Thus, it is very unlikely that significant amount of peracetyl anion exist in these conditions before it self-decompose below its $\mathrm{p}K_\mathrm{a}$. That means it doesn't have a chance to show or form any resonance activity!
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