My hunch is that it would be very difficult without Russian support - the United States can't send people into space without Russian support today, Poland would find it harder to go it alone. I assume you're talking about sending people into space rather than just satellites.
My understanding is that unlike Ukraine, the Soviets didn't station substantial nuclear forces in Poland during the Cold War. If they had done so you could imagine a modest Polish satellite launch programme growing from the remnants of the Soviet Union's missile force left behind in Poland after they withdrew in 1992.
Perhaps Lech Walesa agrees to let Russia build a substantial satellite launch facility in Poland - I have no idea why the Russians would do this, let's imagine that the soil in Poland has special magnetic properties that make it easier to launch satellites. Or that the Russian rockets are incredibly unsafe, so the Russians want to get them off Russian soil. It would be staffed with Polish workers, equals lots of jobs, in exchange for agreeing not to join NATO for twenty-five years, or for allowing Soviet troops in East Germany free passage through Poland back to the crumbling USSR.
In which case it would be a Russian/Polish facility, unless the final collapse of the USSR leaves the plant abandoned to Poland. Fast-forward to today and Poland is now a major component of the European Space Agency (in our world they joined in 2012).
As with everything the key problem is the economy. If Poland found a massive oil reservoir under Poland, or incontrovertible evidence of massive oil deposits on the Moon - that no-one else was aware of - Poland may well now into space.
Or perhaps while growing up Lech Walesa is a massive fan of science fiction, and decides that Poland needs to make its mark on the world with satellites. The problem with this idea is that Walesa would have grown up with Eastern European science fiction, which was grim and existential, so as an adult he would probably be terrified of the unknowable mysteries of space and would hate the thought of going there.
I've just discovered that there really is a Polish space agency. It's called POLSA. It was founded in 2014. "Aspiration of the Agency is that in 2030 Polish space sector will be in selected areas fully competitive on a global scale. We should be able to ensure the independence of Poland in the access and use of satellite data", which almost implies that they want to launch a Polish rival of GPS. In case the Polish army gets lost in Poland, I don't know.
I learn from Wikipedia that Mirosław Hermaszewski is the only Polish person to have gone into space. He spent seven days in space on board Soyuz 30 in 1979. He shared the station with Pyotr Klimuk of Belarus. Imagine if the pair of them had declared independence from the Soviet Union, with Soyuz 30 becoming a new independent state. Resupply would be hard and of course replenishing the population would be extremely difficult, but assuming Klimuk agrees to become a Polish national - and I see no reason why he shouldn't - Poland might already have been in space since 1979. Just one tiny decision prevented that from happening.