Can you send the cops to deport illegal immigrant from their own home?

As a general rule, enforcement of Immigration laws is a Federal matter. Most states do not have laws about illegal immigration. That having been said, if there is doubt that an arrested suspect is here legally then in an increasing number of jurisdiction they will turn the person over to the Feds. But there again, they won't just pick someone up on suspicion. It will be the after effect of an arrest on other charges.

So to your questions:

1) How do you know they're illegals? You had better be REALLY sure because if you're wrong and they find out that you called the cops on them, they could sue you under tort laws.

2) What state do you live in? There may be state laws that allow for this. But a pre-text would be necessary. The cops can't just go up to people and ask them for proof of their citizenship. This isn't the Soviet Union ya know!