They could absolutely get to space. "Rocket science" isn't a complex field of science: it's just ballistics and some straightforward chemistry. What's fiendishly complex is the precision engineering and manufacturing required to go, not just into space, but where you mean to go with very little margin for error.
What this means is that initially your numbers would all be wrong. If you do calculations in a fast-moving environment without taking relativity into account, you end up with numbers that are somewhat different than the true numbers. Your scientists would discover that their rockets, their clocks, everything functioned slightly different than projected.
This sounds bad - a small difference in propulsion is the difference between making it to the moon and crashing into it, or flying off into empty space, never to see Earth again. But the reality is that in space exploration, nothing works quite like you projected it would the first time. The history of space is replete with experiments, tests, and incremental refinement of ideas. The first of anything rarely performed according to the models, but would show us where the models were wrong so we could fix them.
The same would happen with your engineers: they would be able to measure and eventually predict this effect, and would discover that it was tied to the velocity of the object in question. Once they learn to account for the variations, they can run the same calculations that we can, arrive at the same numbers for impulse, velocity, flight time, etc. They can adjust their clocks to deal with minute variations when at speed. Their ability to operate in space won't be impaired.
You could make a fair case that they have discovered relativity then: they can describe perfectly what it does, they just don't know why (or rather, can't describe "why" in relation to their other physical theories). There are plenty of phenomena that we're at this stage with in real life, little things like gravity. It doesn't stop them (or us) from working with these unknowns.