Only root/sudoers have access to docker by default.
Technically, you can:
1) grant the subscription's system user (at Web Hosting Access) with shell access. Then modify user by adding the docker group, for example:
# usermod -aG docker
jdoe
and call the commands in deployment actions afterward.
2) with deployment actions write some content(the commands, for example) to a file, which is checked by some background script(for example, powered with
inotifywait) and executed by background script on behalf of root.
P.S. in both cases, especially the second one you should mind that the whole docker service will be exposed to that user.