My answer is yes, anyone can be a Shinobi.
All humans have chakra. This isn't exactly something you are born with or without. The latest chapter was a bit confusing, but Madara specifically stated that the Sage of Six paths was the first person to "wield" chakra. That means that he was the first person to be able to use it with skill, not the first person born with it.
Chakra is present in every cell of the body
Anyone can be a shinobi if they are taught how to use their chakra. And even for people like Lee who can't use chakra externally for ninjutsu and genjutsu, he is still able to use chakra in other ways, such as walking on water.
As for the last part of your question "without the ability to manipulate it" , I think this part is probably arguable as the definition of a shinobi is arguable. How exactly do we want to define shinobi. For my definition, a shinobi simply just a ninja. Shinobi in Naruto normally take missions for hire and wield chakra.
However, in Naruto, there should not exist people with no inborn chakra and thus anyone can be a shinobi. People aren't born with the ability to manipulate chakra; they learn how to.
EDIT (My additional interpretation of the last chapter):
Here's the way I interpreted it. Everyone contains chakra, but no one knew how to use it. When Kaguya consumed the fruit, she became godlike with excessive amounts of chakra. She then gave birth to a child, who was born into the world knowing how to manipulate it. From there on, the Sage of Six paths taught others how to manipulate it.
Madara said "Chakra was something that originally only belonged to the Shinju" and I think this is just a mythical explanation of how chakra came to humans. Before Kaguya, no one was able to manipulate chakra and therefore chakra only belonged to the Shinju. "By taking the chakra by force" as Madara said, AKA consuming the fruit, humans could figure out how to manipulate it. This also further explains why the Shinju is taking all the chakra back. If for some reason Kishimoto decides to use Madara's interpretation of the Uchiha Tablets as actual history, there will be contradiction to how Kishi defined chakra in the past.
Asian history has a common tendency to include supernatural/mythical features to its stories, like the Romance of the 3 Kingdoms for China. Although there are some certain historical facts to it, stuff like magic is included and it gets all sorts of wacky if we believed everything that was recorded.