Your requirements limit the possible answers to the point of near impossibility. Being able to travel in outer space is basically one of the biggest achievements you can have, and being able to communicate via a well established network of communications is one of the many building blocks necessary to get there.
Why You Need The "Internet"
I'll explain this in detail because some people seem to simply not get it.
A lot of people on this thread seem to be confusing the "internet" with our use of the internet. All the internet is is a set of communication protocols which allow computers to talk to one another. That's it. Whether two computers connect only to one another, or 2 billion servers connect to form the World Wide Web it's still the same concept.
When computers were first invented they were massive things with a single input terminal. They existed only within universities, and military compounds. These various computer centers wanted to communicate with one another. Calling one another and telling the person at the other end what to type into their terminal was not a secure, or particularly great approach, so they developed a way for the computers to communicate directly. Welcome to ARPANET (Advanced Research Projects Agency Network - a military project), aka the forefather of the "internet".
Eventually, due to advances in technology, and the efforts of companies like IBM, Intel, Microsoft, and Apple computers became available to the average person. Having these many computers be able to communicate with one another was useful, so the networking concepts were applied, and the "internet" was born.
People saw the potential which the internet possessed, and moved beyond simple email to start creating websites, etc. Our own psychology and needs shaped the purpose of the World Wide Web into what it is today. Our vanity and narcissism spawned social media, etc.
However, websites, social media, and the like are not what the internet is. They are merely the human interface to it. The content we created because that's what pleases us. No, the internet is the underlying communication protocols which send the information I've just typed to the SE servers, and then to each and every one of your screens.
The reason why any civilization which wants to reach the stars needs an "internet" is because they need a way for all that technology to communicate! Your ship must send data to the ground control center. It must receive information in turn. Ships must also be able to communicate, and transmit information on a level other than the crew radio-messaging each other. And since you want all the different ships to "understand" one another you must develop a standardized way in which they should do so. A set of protocols. That is the internet!
The race in question might not develop social media, or websites because they lack a fundamental interest in such things. Or perhaps they simply lack the imagination. Who cares?
PS: in Star Wars, when Solo and Luke are saving princess Leya R2D2 accesses the Death Star's network and finds out which cell she's being held in. Star Wars totally has networks (aka an "internet").
Aka no computers communicating with one another. Aka stone age. Ok, we can make it work!
The only real option at this point is that your people find the ships on the planet somewhere, and use them without knowing how they actually work.
For example, maybe a mighty race maintained a base on that planet (a giant underground vault containing ships, and even set up a space station in orbit). These were left there in case of emergency, but then that race died, moved on to a parallel universe, or simply forgot about this neck of the universe, and left their technology behind.
After centuries of scratching a living in the dirt your hunter-gatherers stumble into the vault, and the governing AI decides to cooperate with them for some reason (maybe it needs some favors that only these primitive people can provide, or maybe it simply becomes their God and orders them around). It however doesn't give them access to its communication's network (the "internet"), so they are not even aware such a thing exists, or what its potential might be.
They now have the use of the base, orbital station, and ships without understanding the technology. The maintenance is done by the automated systems, and by robots controlled by the AI.