Currently I'm running Ubuntu MATE 16.04 LTS. It has this package in the universe repository.
It is also available in all other Ubuntu versions:
$ rmadison mate-dock-applet
mate-dock-applet | 0.70-1build1 | xenial/universe | source, amd64, arm64, armhf, i386, powerpc, ppc64el, s390x
mate-dock-applet | 0.85-1 | bionic/universe | source, amd64, arm64, armhf, i386, ppc64el, s390x
mate-dock-applet | 0.87-1 | cosmic/universe | source, amd64, arm64, armhf, i386, ppc64el, s390x
mate-dock-applet | 0.87-1 | disco/universe | source, amd64, arm64, armhf, i386, ppc64el, s390x
So you do need any PPA to get it working.
Moreover the ppa:webupd8team/mate
PPA does not support 18.04 LTS (bionic). Consider to remove it from your sources.list
:
sudo add-apt-repository -r ppa:webupd8team/mate
sudo apt-get update
and then simply install the package with
sudo apt-get install mate-dock-applet