You could ungroup audio and video (if they are synced) and cut the word into a single event and use event fx to apply the delay to just that word. Being a creature of habit, I still find myself using the older and more time-consuming method of applying an fx animation envelope to the track and keyframe the fx send on the word I want echoed.
I just realized I am steering you in a totally wrong direction for a delay effect. I had forgotten that Vegas fx engine works differently than my other audio programs. (that's probably why I use the old methods I described earlier) Applying delay as an event fx will cause the fx return to mute at the end of the clip. So your delay will cut off when the cursor passes the end of the clip. The way to do this is to either cut all the words you want to echo into separate events and move them in place to a new audio track, and then apply the Simple Delay as a track fx (the same vertically oriented icon in the track list section at the left of the timeline) or go with Alistair's suggestion of adding an fx buss and using an automation envelope. It may sound complicated the first time you set it up, but once you go through the flow, I think it will make sense and will be something you'll use a lot as you go forward.
I've often created this effect manually by cutting out the word and repeating it by pasting the clip (event) over and over while reducing the volume. Then I add a reverb to the track.... but I also wondered why there isn't a single plug in with a preset that does this.