The other answer has given the major conformer of the protonated form of the o-hydroxybenzoic acid, but I believe there is a part of your question that still remains to be answered. You asked why the hydrogen bond makes the ortho-isomer more acidic than the para-isomer, when the hydrogen bond is already present in the protonated form of the ortho-isomer.
For an explanation of this, let's consider the ionization equilibrium of the ortho-acid:
(I am considering only the major conformer as mentioned in the other answer)
Now, there is hydrogen bond in both the protonated and deprotonated form. But, consider the strength of the hydrogen bond in both cases. In the protonated acid, the H-bond donor is a neutral $\ce{>C=O:}$. In the deprotonated state, the H-bond donor is an oxygen of a carboxyl anion. The negative charge is delocalised over both of the oxygen, so the H-bond donor $\ce{O}$ in this case has some extra electron density compared to the neutral $\ce{>C=O}$. So, it can form a stronger H-bond.
This means the H-bond is stronger in the deprotonated state than in the protonated state of the o-hydroxybenzoic acid. So, the presence of H-bond still makes the ortho-isomer a stronger acid (than the para-isomer where there is no intramolecular hydrogen bond), even if the H-bond can exist in both protonated and deprotonated forms.