Is usage of "leverage" as a verb in "Leverage competencies to the value chains of other existing businesses" correct?

The first sentence is more grammatically correct than the second.

If you are to use the preposition "for," better include the reason for leveraging competencies as "for" is "used to indicate why something is done".

See http://learnersdictionary.com/definition/for

Hope this helps.

The usage of "leverage" in the phrase quoted by you is correct. It has been used in exactly the same sense that you have stated.

The usage of "to" is slightly nonstandard, but it's not wrong. I've seen such usage in scientific journals as well.

The writer probably means that competencies acquired for a purpose can be mapped to the value chain of other businesses. For example, I might acquire a master's degree in Earth Science. However, I might use the same skill sets and knowledge as the Editor of an Earth Science magazine instead of pursuing a career in mainstream Earth Science.

It depends on how you use the content. Example....way to capitalize on your position is by leveraging it. In other words, to leverage (verb) your leverage (noun).

It takes a verb, to lever, that has become a noun, leverage, and twists the word into another verb even though it ends with the noun-defining ending –age.

Lever = (Verb) - Leverage = (Noun)

By adding the suffix (age), it becomes a noun.

Advance = (Verb) - Advantage = (Noun)

It's the same concept as they say if you change an ingredient on a recipe, that becomes a new recipe.

Leverage is not a verb nor should it be used as one. The English language is rich enough to contain plenty of verbs to indicate the movement required. Manipulate, twist, manoeuvre, move, lever, push will all fill the need.