Me toca
 

... lo que pasa es que me toca de todas maneras llamar a la policía.

This is translated as "... I have to call the police anyway"

The dictionary suggests toca is either second or third person - Is this is an imperative of first person as well?

I would expect to hear "tengo que" or similar, is toca and equivalent term?

There's no such thing as a first-person imperative, in the singular form.

'Me toca' can be literally translated 'it touches me'.
In this case, the verb is being used in the third-person ('it' being the unwritten/unspoken subject). The verb is not being used as an imperative.

When it comes to translating 'me toca' into English, you'll often use "it's my turn," but in this setting it takes on more of a personal 'have/must-do' idea.

By the way, the imperative 'touch me' is 'tócame', in the second-person conjugation. The third-person conjugation, used when the subject is the formal second person (usted), is 'tóqueme'.

'Tocarle hacer algo' is the model being used in the sentence, and is the equivalent of 'tener que hacer algo', as you suspected.

A translation which captures the sense of external obligation here would be "it falls to me to call the police".

Thank you Rusty very comprehensive

Thank you that's great

Do you mean that I have to call the police because you called it las time?

No, it's not about turn-taking, but it does suggest an element of doing something out of duty rather than because you want to.

To build on what pjt33 is saying, other good options might be:

It's up to me to call the police.
Calling the police is on me.
It's on me to call the police.