Learn French: Il faut que nous signions la pétition pour le maire.

It is actually the subjunctive form, required here by the expression of necessity Il faut que (we must / it is necessary that). For regular ER verbs like signer, the nous and vous forms in the present subjunctive are identical to their imperfect forms.

The English translation is 'We must sign', so why doesn't the French sentence start with Nous?

Il faut que is an impersonal expression that literally means 'It is necessary that'. While you could say Nous devons signer using the verb devoir (must), French speakers use Il faut que plus the subjunctive all the time. It is just a different structure where the 'we' (nous) becomes the subject of the second part of the sentence.

Could I say Il faut nous signer instead to avoid the subjunctive?

No, because Il faut nous signer would mean 'It is necessary to sign us' (treating nous as a direct object receiving the action). If you want to avoid the subjunctive, you could say Nous devons signer (We must sign) or you can drop the specific person entirely if the context is obvious: Il faut signer (It is necessary to sign).

What if I wanted to say 'The mayor must sign the petition'?