Delay - need guidance

Normally I automate the delay in situations like this. Usually there is some sort of 'input' value which you can switch off most of the time and just raise the value at the end of a phrase. You can probably manually draw in the envelope yourself in an automation lane. You may need to automate the delay output level (or the bus output level) as well if the delay sometimes repeats for too long.
 
Some delays come with built in 'ducking' capability and while that does help if used live, I don't think it's an adequate replacement for manually automating the input and output to the delay when working in a DAW.

Volume automation on the bus. You'll want to bring the delay in gradually, over a quarter- to a half-second, to avoid an abrupt change. It's the kind of thing you want to draw in by hand and audition, rather than recording the automation.
 
Check to make sure the delay you're using doesn't have a wet/dry mix knob, because if it does you can just automate that control, eliminating the need for a separate bus.

Having the delay on the buss which the vocal track outputs to is no different from having the delay as an insert effect on the track.
You might be better off running the send to it's own bus in a send/return configuration (especially if you want more than just one vocal track to be effected) and bring the delay up in parallel. Automate the bus volume to bring the effect in and out, or automate the send levels on individual tracks.

See "Craig Anderton's - Tip Of The Week No.64". This shows 'gating' of a reverb on a vocal track. Just swap the reverb for a delay and you have what you're after. I think Craig's tips are great, and often find things there that are a genuine help in my everyday workflow.