Does the technology exist? Yes, and it has since it was clear that docking was reliable enough to be used in the Apollo project. When the Lunar Module ascended from the Moon, it docked with the CSM. That is by most common measures to dock with a booster. After all, the main purpose of the SM was to insert the Apollo stack into lunar orbit, and then depart towards Earth. That is very much a booster role. Through hundreds of visits to space stations, docking is made an everyday and safe operation. We have the technology.
Does the spacecraft exist? No, not at the moment. While for example space station resupply craft, like Progress, have engines, and can be used for station re-boost, they are not dedicated boosters by themselves. Apart from space stations, docking is not much used. Today, we have no such boosters. Building them is however not an issue if needed.
The scenario you describe is very close to the concept of the Parom space tug, a planned component for Clipper, but may also be used for in-orbit cargo transportation. We have yet to see how many of those plans will be realized.
As a historical reference, the Agena target vehicle used by Gemini, was among other things also used for propulsion, for example for setting the at-the-time record for being farthest away from Earth.