Is it a common practice for political pollsters to use random independent samples to conduct their polls?
You have asked two questions. Your title question asks only about independence. Yes. Polling has lots of problems but getting independent samples is not usually one of them.
But in the body of the question, you ask about "random independent samples" and that's much trickier. It will depend on what the poll is about, but perhaps the most common polling is about politics and here it is pretty much impossible to get a random sample of voters - because there's no list of voters. There may be a list of people eligible to vote, but not everyone who is eligible winds up voting and the people who vote are not a random sample of those who are eligible.
In addition, the people who answer the pollster are not a random sample of those who they call. So, even if the pollster could get a list of everyone who will vote, they'd have a problem with the people who answer not being a random selection.
Political pollsters spend a lot of effort working on this. The usual method is to weight the sample of respondents to match the people they think will wind up voting. They base this on past elections and, sometimes, on their own thoughts about how this election will be different.
That's one reason that different polls, all with reasonable sample sizes, taken on the same day, can very different results. It's also the main reason that polls can be very wrong.