It sounds very much like the vacuum booster on the brake master cylinder has a ripped diagram internally. That is the black round unit that the vacuum line plugs into.

With the engine running have someone step hard on the brakes and listen closely in the area for a continuous hissing sound. If you have a hiss the diagram is leaking.

And if it is leaking it will also cause the engine stumble. One way to verify that is pull the vacuum line off the booster and plug the line. If engine stumble is gone you found the culprit.

1998 Sportage - Gone.
2004 Honda Civic Coupe.
2007 Suzuki Grand Vitara - She got it.