Adding a water storage tank

Yes you can. A combi boiler works the same as a system boiler but with the benefit of instant hot water.
Simply by adding a zone valve and a separate timer this can be easily done.
Zone valve would Combine with the heating output and divert to the tank.

On the hot water side of things you would have to install a check valve to stop the high pressure hot water flowing back into the tank if you plan to join to the existing hot water pipework. If you kept the hot water from the tank separate like to just the shower or bath then this would not be needed.
Or fit a pressure reducing valve to the cold water entering the boiler, or just close of the hot water from the boiler and just use the setup from the tank.

You need someone that clued up to separate water tanks and boilers. Then a normal plumber that only knows about combis. Don't trust the fact someone is gas safe they are competent as gas safe is only to do with gas and unfortunately most aren't clued up to the in and outs of plumbing. (edited)

Here is a video of a unvented of a combi. Ignore majority of the video as it's to do with adding a weather compensator to the system.