Yes, both systems use vacuum from the vacuum pump mounted on the front of the engine.
Since your car has so few miles I would guess that the pump has swallowed its check valve. The valve is available separately MB# 004 997 18 72 $21.50 list.
To check first determine if lack of source vacuum is your problem. Next shut-off motor and remove the hard plastic hose from the pump to booster. Remove at the pump only. Beneath the line is a removable fitting 19mm if I recall. This fitting is the check valve. Remove it and you shouldn't be able to see through it. The guts are inside the pump. You should get the rest of the pieces out by removing the front plate. Or, you could replace the whole pump MB# 000 230 30 65 for $410 list (partsshop could really help here I bet).
Owner 24 bay BSC
Bosch Master, ASE master L1
26 years MB technician