Assuming their metabolism isn't far higher than their size would suggest (it has to be higher than a human's, but they ought not to need more food, water, and oxygen than a house cat at that size), they'll find it much easier than we did.
The first humans into orbit required multiple tonnes of life support and thermal protection equipment, equipment with size and mass determined by the size and mass and metabolism of an adult human. Your one-foot humans wouldn't need anything like as much space or as much oxygen/water/food per hour (even though it'll be more consumables per mass of astronauts because small creatures have higher metabolisms).
Compare Sputnik 2, which carried a dog named Laika into orbit (the first mammal to orbit the Earth): it was only about twice the mass of Sputnik 1, the very first artificial satellite. Gagarin's capsule, by contrast, was more than fifty times as heavy. And Laika was significantly heavier than your little people.
They need to carry only a fraction of the support and needing a lot less space means their capsule can weigh a lot less. Therefore, their stand-in for a Mercury spacecraft, which required a second-generation Atlas booster to reach orbit, would weigh no more than a tenth as much -- which means the rocket that launches it can also weigh a tenth as much.
To put it in technological stages, humans going into orbit had to wait until kerosene-burning rocket engines were available -- but your little people can go into orbit with two- or three-stage alcohol burners, which means they can (if they choose) go, relatively speaking, about a decade before we did -- and start building space stations with the launchers we used to put the first people into orbit.