I’m guessing the same way the Russians sent a dog in space, i.e. Rockets!!! (The first animal to actually go into orbit was the dog Laika, launched on board the Soviet Sputnik 2 spacecraft on November 3, 1957 – Wikipedia look it up 🙂 )
You need a really tall tree with very low hysteresis (i.e. very springy). You bend the top of the tree down to near the ground, and tie it down with some rope, then you set a bear trap which grabs the bear and releases the rope, the tree will spring up straight again and throw the bear into space. If we say that space is 100km up, and it’s a black bear with a mass of about 100kg, then you’re going to need to release about 100MJ of energy, which is quite a lot to store in a bent tree. Also, that’ll only get the bear into space briefly before it falls back down again, so you’ll need a large trampoline to catch it unharmed.