It is entirely possible to reach Elite in Exploration without ever leaving the Bubble, just by honking and FSSing every star system you visit and by using the DSS probes to Map every planet you come close enough to. It might take much longer than doing it by actual exploring, but you can do it. There are many inhabited Earth-like planets in the Bubble that you are likely to visit just by flying around randomly doing regular trade, combat and misisons, and Mapping these planets as you pass by them earns you a few hundred thousand credits, each. That may not be much cash for a grizzled veteran with a fleet-carrier-sized budget, but for a newbie, you can easily get enough cash just by Mapping a single inhabaited Earth-like or two to buy a whole new ship and fully upgrade it.
You should be aware that there are two different types of inhabited star systems: "Explored" and "Unexplored". Explored systems typically have populations over 1 million, and you do not get credits for, nor have a need to, run the FSS scanner in such systems; if you honk you will notice that the entire system has already been pre-scanned with all planetary information available to you without the need for FSSing. You can still get credits for Mapping the planets in such systems, with a few exceptions (the Superpower capitals of Sol, Achenar and Alioth, for example; you cannot Map their worlds). Planets in such systems will show a "Mapped by" Tag, but no "First Discovered By" Tag.
"Unexplored" inhabited systems usually have a population under 1 million. These are the systems that require you to run the full FSS scan in order to see planetary details. The planets will show on the system map as "Unexplored" until you either run the FSS or you fly to within 32 Ls of the planet. Planets in such systems have both "discovered by" and "mapped by" Tags. You do get paid credits for honking and scanning these systems, just like you would in a truly "unexplored" system - only you won't get the Tags and First Discovery bonuses.
It may seem odd for the game to classify a star system that's home to hundreds of thousands of people as "Unexplored", but this is how it is. It was designed that way so that newbies can still get some sense of "exploration". If you actually had to travel thousands of LY before you started to get "exploration credit", I doubt many new players would even attempt to begin the exploration pathway.