Hello Ashley,
As Susanne says, there is an equivalent in Spanish for that expression: "estar de guardia".
You can be "de guardia" ("on call") either at work or out of work, being alert to your phone.
On call jobs are not rare in the Spanish speaking world; they are actually almost as common as in the US or Canada: doctors, nurses, judges, policemen, firemen, entrepreneurs, etc, they all have on call jobs. The reason why Spanish speakers use the English expression when they talk is because we usually don't say "estoy de guardia" when you are out of work, but "estoy pendiente del teléfono" (I'm on the lookout for any calls) or "me pueden llamar en cualquier momento" (I can be reached anytime).
When immersed in a non-Spanish speaking country, you adopt, as it is logical, some of the expressions of the hosting country, specially those that offer you shortcuts for communicating. And "on call" is one of them.
I hope this helps!