What is the adjectival form of "initiative"?

Somewhat to my chagrin, I think I'd go with proactive.

The adjective usage of initiative that you're finding documented would have to be spoken as "inish-ee-aytiv" (as opposed to the noun "inish-uh-tiv") in order to be understood, it would sound distractingly odd, and your chances of getting your meaning across still aren't too hot. In written communication it would be a total loss.

In some circumstances you could go with a fairly close mapping to initiatory, but that would not typically be understood properly either.

The OED also records initiative as an adjective, but I’d be surprised to find it used as such today. In your example, I think you’d have to use a different word altogether, such as enterprising.

Someone with ( or who takes ) initiative can be described as motivated or ambitious. Motivated is the less intense of the two, meaning that the person wants to get things done, while ambition ( a more intense synonym of initiative ) can reach the point where it is "unhealthy"; for instance, a business partnership could be harmed by one of the partners being so ambitious that they cut the other out of the business partly or even entirely.

When someone is so ambitious that they are willing to sabotage relationships, we can describe them as cutthroat. This adjective can also be applied to situations, eg. "the computer hardware business is cutthroat" ).