"take the initiative" vs "make the initiative"

The phrase make the initiative is not grammatically incorrect. Make is a transitive verb and initiative is a noun that logically can be made.

However, take the initiative is an idiom that is quite common in English. Make the initiative is effectively never used. Here is an ngram (a chart reflecting the existence of terms in a large database of literature) that shows the relative frequency of the two phrases.

While it would not be formally wrong to use make . . . it would be considered odd by most listeners who were fluent in English.

I've never heard "make the initiative". I'm not sure it's valid grammar.

Edit:

Ah I found a valid use. An article used it like this: "there wasn’t a way to make the initiative work."

Which is a totally different context than taking the initiative; its talking about the initiative being a plan the government has.

The only other usage I can find is people on yahoo answers asking whether they should "make the initiative" to ask someone out, and I wouldn't trust their grammar at all. They should've said "take".

The Ngram chart for "take the initiative" (blue line) versus "make the initiative" (red line) for the period 1750–2005 shows that neither phrase was at all common in English until the early 1800s, when "take the initiative" began a long ascent in frequency that lasted until the 1970s:

There are enough instances of "make the initiative" to permit Ngram to track it, but a substantial number of those instances involve formulations such as "make the initiative a success," "make the initiative more palatable," and "make the initiative viable." The Ngram chart for "take the initiative to" (blue line) versus "make the initiative to" (red line) for 1750–2005 doesn't include a red line:

because, according to Google, "Ngrams not found: make the initiative to." Since the context of the usage you have in mind is whether you should "take the initiative to contact X" or "make the initiative to contact X," your example falls into the "take/make the initiative to" word string—and for that string, Google Books finds many instances of take and no verifiable instances of make.