What was really important to our development of technology was not oil, but coal. Access to large deposits of high-quality coal largely fueled the industrial revolution, and it was the industrial revolution that really got us on the first rungs of the technological ladder.
Oil is a fantastic fuel for an advanced civilisation, but it's not essential. Indeed, I would argue that our ability to dig oil out of the ground is a crutch, one that we should have discarded long ago. The reason oil is so essential to us today is that all our infrastructure is based on it, but if we'd never had oil we could still have built a similar infrastructure. Solar power was first displayed to the public in 1878. Wind power has been used for centuries. Hydroelectric power is just a modification of the same technology as wind power.
Without oil, a civilisation in the industrial age would certainly be able to progress and advance to the space age. Perhaps not as quickly as we did, but probably more sustainably.
Without coal, though...that's another matter
You need some abundant, high-density energy source to bootstrap your civilisation up to the point where renewable energy sources can take over. We used coal for this, so I'll use 'coal' as a stand-in for 'abundant, high-density energy source'.
If you can't smelt steel or copper in large quantities, you can't build wind or hydroelectric turbines, you can't distribute the power, and you can't release all the labour that goes into farming for everybody.
You could use charcoal for a while, but it's horribly inefficient compared to coal. Think of the beautiful, wind-swept, unspoiled hills and moors of places like Scotland or Wales. They're famously unspoiled and untouched...
Except that they're not. Those landscapes are entirely man-made. Those hills used to be absolutely covered in conifer forests, but the trees were cut back, in large part to be turned into charcoal. If you tried to fuel an industrial revolution on charcoal...you might be able to manage it, with laser-focus on the goal of getting to renewables, and probably with an entire continent of trees being harvested, but by the end your economy would be teetering on the brink, and it'd be touch and go whether you and your tech could survive.
So the critical component is not oil, but coal. You need coal (or something like it) to get your tech base up to the level of using other energy sources - like wind, solar, geothermal, nuclear, etc - on an industrial scale, and you need that tech level to be a space-faring civilisation. If our civilisation collapsed today back to a medieval level, it's questionable whether we'd be able to rise again, because so much of our easily-accessible high-quality coal has been used up.