What's the meaning of "take their word for it"?

Question

From NPR, Charleston Mayor: Company Behind Chemical Leak Run By 'Renegades,

It was an interview with the mayor of Charleston, W.Va., about the city's water supply that had been contaminated. The following is what Danny Jones, the mayor, said after being asked "But how do you convince people that the water supply is, in fact, safe?":

"People want to drink water and people want to bathe in it. People want to use it to cook with and they're looking to take the word. And there have been a lot of precautions. We have a very capable health director here in the county. He's watching this very closely. This water is being tested by the hour, by the day. And I believe if they say to go ahead, I would take their word for it and I think the public, at large, will, too."

What's the meaning of the two variations of take their word for it above?

Answer 1

As Nick Stauner's explained in his answer, both "take the word" was used to refer to "to accept their word for a conclusion of the issue." However, it might be beneficial to consider each of the involved sentences, one at a time. Here is the first one.

People want to use it to cook with and they're looking to take the word.

This is obviously related to the question "But how do you convince people that the water supply is, in fact, safe?". It means people want to use the water, and they want to be assured that the "water is safe".

We can understand this take the word by looking up the words take and word in a dictionary. For example, here are the related entries from Macmillan Dictionary,

take _{5 [transitive]} to accept something that someone offers you
word _{4 [singular/uncountable]} news or information about someone or something

_{Thus, we can understand "they're looking to take the word" as the people are looking to accept the news or information (in this case, the approval from the Health Department, according to a preceding paragraph in the transcript). Also note that this is not a common usage, as J.R. and others commented. Take someone's word for it is more idiomatic and recommended for learners.}

Here is the second sentence,

And I believe if they say to go ahead, I would take their word for it and I think the public, at large, will, too.

The phrase take someone's word for it is common enough that dictionaries would list it. Here is what I found in the same dictionary,

take _5a. to accept an explanation or something that someone says without discussing it or arguing about it
  take someone's word for it (=believe what someone tells you): You don't have to take my word for it – ask anyone.

_{This means that the mayor trust the Health Department, and if they say the water is safe, he would believe them.}

Answer 2

The first instance is a little unclear until reading further; I'm guessing you've omitted some important context immediately preceding this quote. In the second instance, if the water testers "say to go ahead," the implied "word" would probably be their statement that the water is potable (safe to drink). To "take their word for it" is to trust them, or more directly, to "accept their word for a conclusion of the issue." It seems safe enough to assume that the first instance refers to the same word as the second instance, given the additional context following the second sentence.