Is the sentence “It provides people an easy way to communicate.” grammatically correct?

"Provide" has two different subcategorisation frames:

provide somebody with something

and

provide something [for somebody].

In the latter structure, it can (like "give" and "show") be transformed to "provide somebody something".

So both forms are grammatical and I don't find a difference in meaning betwen them.

I think if you include with, it might change the subject of "people" to "people with an easy way to communicate" as if the subject is changed not only to people, but people who can communicate easily if that makes sense. The point of your sentence is to provide people an easy way to communicate. So I think keep it the way it is.

The ‘Cambridge Grammar of English’ (not to be confused with ‘The Cambridge Grammar of the English Language’) is primarily for non-native speakers, but is nonetheless authoritative. Of verbs like provide it says they ‘have special prepositions associated with them and are only used in the oblique construction, not with indirect and direct objects.’

There are several basically equivalent ways to use provide:

provide (someone) with (something)

provide (something) to/for (someone)

provide (someone) (something)

The last is less common than the other two, but it’s not hard to find, as these sentences from the Corpus of Contemporary American English attest:

That’s the choice you’ll have is having your employer no longer provide you health care. (spoken)

A few, like the Gordons’, were partially and crudely roofed to provide them some shelter and shade from the summer sun. (fiction)

…the present findings do provide us some insights in explaining the specific circumstances under which accountability will influence cooperation… (academic)

A Google search for “provide us a” gets millions of hits, including this quote from Scott McCloud’s TED talk about comics:

I think this is important because media, all media, provide us a window back into our world.