The one-word buildup is only a noun or an adjective. It does not work as a verb. When you need a verb meaning to develop by degrees, to accumulate, or to bolster, use the two word phrasal verb build up.
Some publications, especially British ones, favor of the hyphenated build-up instead of buildup. But compound words like this almost always lose the hyphen in the end. Some words take a little longer than others.
Although build-up has some proponents, buildup appears throughout the English-speaking world—for example:
A buildup of gas in a reservoir drop shaft might have been the cause of an explosion. [Chicago Tribune]
Britain has kicked off a global advertising campaign as the international buildup to the 2012 Olympic Games in London continues. [Independent]
Then when Taylor did return to the boat after his surgery, the truncated buildup meant they lost precision. [New Zealand Herald]
And build up is the correct term when you need a verb—for example:
As Mississauga tries to build up its downtown from scratch, it’s adding two more parks and a campus of Sheridan College. [Globe and Mail]
The same thing happens with supersonic jets. As they fly faster than the speed of sound, they build up a shock wave in front of them. [Wired News]
But its efforts to build up its stake are being frustrated by the jewellery company Warren James. [Guardian]